Evaluating water reuse applications under uncertainty: a novel picture fuzzy multi criteria decision making medthod

Published Online November 2018 in MECS DOI: 10.5815/ijieeb.2018.06.04

Reuse of water refers to the treatment and rehabilitation of non-traditional or deteriorated water for beneficial purposes [10]. Water reuse is synonymous with using reclaimed water, which can provide an option to reduce water scarcity, especially under the new reality of climate change and the increase in human activities. Water reuse has become widespread all over the world to solve the depletion of water resources, leading to the reduced water supply. Evaluation of water reuse applications is a weight replacement process and the most appropriate selection of water reuse applications. From this, the assessment involves analyzing many criteria with social, technical, economic, political, environmental and technical aspects to ensure sustainable decision making [21]. The challenge with water reuse application evaluation (WRAE) is that alternatives are diverse in nature, and often have conflicting criteria. The fuzzy set theory [20] is a very effective method for solving such contradictory and uncertain problems.

Fuzzy set theory was introduced in 1965 by L. A. Zadeh. Immediately, it became a useful method to study the problems of imprecision and uncertainty. Since then, a lot of new theories treating imprecision and uncertainty have been introduced. For instance, intuitionistic fuzzy set was introduced in 1986, which is a generalization of the notion of a fuzzy set. While fuzzy set gives the degree of membership of an element in a given set, intuitionistic fuzzy set gives a degree of membership and a degree of non-membership. Picture fuzzy set [4] is an extension of the crisp set, fuzzy set and intuitionistic set. A picture fuzzy set has three memberships: a degree of positive membership, a degree of negative membership, and a degree of neutral membership of an element in this set. This approach is widely used by researchers in both theory and application. Dinh et al. investigate the distance and dissimilarity measure on picture fuzzy set and apply it in pattern recognition [13]. Hoa and Thong (2017) improved fuzzy clustering algorithms using picture fuzzy sets and applications for geographic data clustering [11]. Son [7, 8] given an application of picture fuzzy set in the problems of clustering. Dinh et al. introduced the picture fuzzy database and some examples on picture fuzzy database [12].

We often use decision-making methods because of the uncertainty and complexity of the nature of decisionmaking. By the MCDM methods, we can determine the best alternative from multiple alternatives with respect to some criteria. In recent times, the choice of suppliers has increasingly played an important role in both academia and industry. Therefore, there are many MCDM techniques developed for the supplier selection supplier selection [2, 3, 5, 9, 19, 22]. However, the above methods are limited to use in set theory. Therefore, it is difficult to encounter problems of uncertain or incomplete data. There are some authors who have proposed MCDM methods using fuzzy set theory or intuitionistic fuzzy set for the supplier selection [4, 6, 9, 14 ,16, 17, 18,21].

With the considered criteria for water reuse options, there are usually three levels. For example, the public acceptability attribute has three levels: agreement, disagreement, and neutrality; freshwater saving has also three levels: low, middle, and high. Therefore, we use the multi criteria decision making method based on picture fuzzy set to select the best alternative in evaluating water reuse applications. The rest of paper is organized as follows: section 2, we recall the concept of picture fuzzy set and some operators of two picture fuzzy sets. In section 3, we propose a new MCDM method using the similarity measure of picture fuzzy sets. Finally, we apply the proposed method for evaluating water reuse application in section 4.

• II.    Preliminarity

In this section, we recall the definition of picture fuzzy sets and dissimilarity measure of them. After that, we introduce a new dissimilarity measure of picture fuzzy sets.

• A.    Picture fuzzy sets

Here we recall the concept of picture fuzzy sets.

Definition 1 [4]. Let U be a universal set. A Picture fuzzy set (PFS)     A on the U is

A = { ( u , m_a ( u) n _A ( u ), Y a ( u ) ) I u ^e U } where M _a ( u ) is called the “degree of positive membership of u in A ”, n ( u ) is called the “degree of neutral membership of u in A ” and y_a ( u ) is called the “degree of negative membership of   u in   A  ” where

ЦА (u), n (u), Ya (u) e [0,1]   satisfy  the following condition:

0 <  m _a (u ) + n _A ( u ) + Y A ( u ) <  1, u g U .

The family of all picture fuzzy set in U is denoted by PFS ( U ) .

Definition 2. The picture fuzzy set B = { ( u , M ( u)n_B ( u ), Yb ( u ) ) I u G U } is called the subset      of      the      picture      fuzzy      set

A = { ( u , M a ( u ), П а ( u ), Y a ( u ) ) I u g U }               iff

M b ^{( u} ) <  M a ^{( u} ) П в ^{( u} ) <  П а ^{( u} ) and Y b ^{( u} ) >  Y a ^{( u} )

for all u G U .

• B.    Dissimilarity measure of picture fuzzy sets

In this section, we recall concept of dissimilarity measure for picture fuzzy sets.

Definition 3. Given U = { u , u ,..., u_n } is an universe set. For any A , B G PFS(U ) . A function Dis : PFS(U ) x PFS(U ) ^ [0,1] is a dissimilarity measure between PFS-sets if it satisfies follow properties:

pf-d iss 1: Dis ( A , B ) = Dis ( B , A ) ;

PF-Diss 2: Dis ( A , A ) = 0

PF-Diss 3: If A C B C C then

Dis ( A , C ) >  max { Dis ( A , B ), Dis ( B , C ) } .

Now, we can identify some new dissimilarity measures between picture fuzzy sets.

Given U = { u , u ,..., u „} is an universe set. For any A , B g PFS(U ), we denote

TA (ui) = Ma (ui) + Па (ui) - Ya (u)

TB (ui) = Mb (ui) + Пв (ui) - Yb (ui)

Sa (ui) = Па (ui) - Ya (ui)

Sb (ui) = Пв (ui) - Yb (ui)

and nMm  । Ta (u) — Ta (u ) + SA (u) — Sa (ui )

Di(A,B) =----------------- for all i = 1,2,..., n .

Definition 4 . Given U = { u , u ₂,..., u_n } is a universal set. For any A , B G PFS(U ) , a dissimilarity measure Dis : PFS(U ) x PFS(U ) ^ [0,1] is defined by

DM t ( A , B ) = ¹ jh D i ( A , B )       (6)

n i = 1

Theorem 1 . DM ( A , B ) in the eq.(6) is a dissimilarity measure on PFS ( U ) .

Proof.

From eq.(1), eq.(2), eq.(3) and eq.(4), we have

• - 1 < T a ( u i ), T b ( u i ), S a ( u i ), S b ( u i ) <  1

for all i = 1,2,..., n .

Hence, according to Eq.(5) we have D ( A , B ) G [0,1] .

So that, from Eq.(6) we have DM_T ( A , B ) g [0,1] for all A , B g PFS ( U ).

(PF-Diss 1). It is obviously.

(PF-Diss 2). It is obviously.

(PF-Diss 3). If A c B C C then

M a ⁽ u i ) <  M b ⁽ u i ) <  M c ⁽ u i )

‘ П а ⁽ u i ) <  П в ⁽ u i ) <  П с ⁽ u)

Y_A ^{( u} i ) >  Y b ⁽ u i ) >  Y c ⁽ u i )

for all u G U .

So that T a ( u i ) <  T b ( u i ) <  T_c ( u i )

and

Sa(ui) < Sb(ui) < Sc(ui)

| T a ( u.) - T c ( u i ) >  max {| T a ( u i ) - T b ( u i ), T B ( u i ) - T c ( u,)\ } and

I ^S A ⁽ u. ) - ^S C ⁽ u )| >  max {| ^S A ⁽ u ) - ^S B ⁽ u )|, I S B ⁽ u ) - ^S C ⁽ u )| } Hence DM_T ( A , C ) >  max { DM_T ( A , B ), DM_T ( B , C ) } .

It means PF-Diss 3 satisfy. □

Now, we assign U_t is the weight T , such that

Y щ = 1. We can define a dissimilarity measure of two i = 1

picture fuzzy sets as follows:

Definition  5.  Given U = {u, U2 ,..., Un}  is a universal set. For any A, B G PFS(U), a dissimilarity measure Dis : PFS(U) x PFS(U) ^ [0,1] is defined by

n

DM T ( A , B ) = E ® i D i ( A , B )       (7)

i = 1

Definition 6. Given U = { U, , U ₂,..., U_n } is a universal set. For any A , B G PFS(U ) , a dissimilarity measure Dis : PFS(U ) x PFS(U ) ^ [0,1] is defined by

n

DM T ( A , B ) = E T i D i ( A , B )        (8)

i = 1

Where

_D,P ^n, . T A ( U i ) - T A ( U i ) P +1 S a ( U i ) - S A ( U i )| P   ₍₉₎

for all i = 1,2,...,n, p G N*

Theorem 2 Given U = { U_v , U ₂,..., U_n } is a universal set.

For any A , B G PFS(U ) . We have

• a)   DM ^T ( A , B ) = ¹ Y TDi ( A , B )

n l = 1

• b) DM T ( A , B ) = ¹ £ TD P ( A , B )

n i = 1

are the dissimilarity measure between picture fuzzy sets. Proof.

• a)    We have

From eq.(1), eq.(2), eq.(3) and eq.(4), we have

• -1 < Ta(Ui),Tb(Ui),Sa(ui),Sb(Ui) < 1for all i = 1,2,..., n.

Hence, according to Eq.(5) we have D. ( A , B ) g [0,1] .

So that, from Eq.(7) we have DM T ( A , B ) g [0,1] for all A , B g PFS ( U ).

(PF-Diss 1). It is obviously.

(PF-Diss 2). It is obviously.

(PF-Diss 3). If A c B c C then

Pa(Ui) < Pb(Ui) < Pc(Ui)

‘ П а ^{( U} i ) <  П в ⁽ ui ) <  П с ^{( U} i )

Ya ^{( U} i ) >  Y b ^{( U} i ) >  Y c ^{( U} i )

for all U G U .

So that Ta(Ui) < Tb(Ui) < Tc(Ui)

and

Sa (Ui) < Sb (ui) < Sc (Ui)

| T a (u,) - T c ( u , )| >  max {| T a ( u , ) - T b ( u , )|, T B ( U ) - T c ( U ,)\ }

| Sa ( U , ) - S c ( U ,)\ >  max {| S a ( U , ) - S b ( U , )|,| S b ( u i ) - S c ( u i ) } Hence DM T ( A , C ) >  max { DM T ( A , B ), DM T ( B , C ) } .

It means PF-Diss 3 satisfy.

• b)    We have

From eq.(1), eq.(2), eq.(3) and eq.(4), we have

• -1 < Ta (Ui), Tb(Ui), Sa (Ui), Sb(Ui) < 1for all i = 1,2,...,n .

Hence, according to Eq.(5) we have D ( A , B ) G [0,1] .

So that, from Eq.(7) we have DM T ( A , B ) G [0,1] for all A , B G PFS(U ) .

(PF-Diss 1). It is obviously.

(PF-Diss 2). It is obviously.

(PF-Diss 3). If A c B c C then

'Pa ^{( U} i ) <  P b ^{( U} i ) <  P c ^{( U} i )

^ Па(Ui) < Пв(U) < Пс(Ui)Ya(P) > Yb(Ui) > Yc(Ui)

for all U _; G U .

So that

Ta (Ui) < Tb (Ui) < Tc (Ui)

and

Sa(Ui) < Sb(Ui) < Sc(Ui)

| Ta(u.) - T c ( u, )| >  max {| Ta ( u, ) - Tb ( u, )|, T B ( u, ) - T c (, u,)\ }

| Sa ( U,) - S c ( U,)\ >  max {| S a ( U,) - S b ( U,)\, S B ( u,) - S c ( U i ) } Hence DM ^T ( A , C ) >  max { DM ^T ( A , B ), DM ^T ( B , C ) } .

It means PF-Diss 3 satisfy. □

• III.    The Proposed Mcdm Method

  C j

  
  

  In this section, we propose a new method for the multicriteria decision making problems which based on the dissimilarity of picture fuzzy sets. The multi-criteria decision making problem is determined the best alternative from the concepts of the compromise solution. The best compromise solution is the alternative which obtains the farthest dissimilarity measure from each alternative to the perfect choice. The procedures of the proposed method can be expressed as follows. Let A = { A , A ,..., A_m } be the set of alternatives. Let C = { C , C ₂,..., C_n } be the set of criteria with the weights of each criteria Cy is A where j = 1,2,..., n

  
  

  i

  d i, = -ci— , i = 1,2,3

  ■ z c i

  i = 1

  
  
  
  

  A 1

  A 2

  
  

  ...

  A m

  
  

  and

  
  

  n

  Z A = 1 . The picture j = 1

  
  

  fuzzy decision making

  
  

  ^d 1¹ j d 2 j

  
  

  ...

  d ¹ mj

  
  

  ^d 1² j

  ^d 2² j

  
  

  d ² mj

  
  

  Then decision making matrix D =

  
  

  ^d 1³ j

  2 j

  
  

  d ³ mj

  
  
  
  

  |^ dy J , in which

  
  

  dy = ( d1, d _z2 , d³ ) G PFS ( C ) , is a picture fuzzy decision making matrix, where j = 1,2,..., n and i = 1,2,..., m . It means that A is a picture fuzzy set on

  
  

  the criteria set C = { C , C ₂,

  
  

  ...,

  
  

  C n } .

  
  

matrix D = | dyJ  in which  dy =( dV, dz2, d3)G PFS(C) , where j = 1,2,...,n and i = 1,2,...,m as follow

	C 1	C    ..	.   C n
A 1	d 11	d .	..    d 1 n
A 2	d 21	d .	..     d 1 n
... A m	... d m 1	...       . d m 2 .	..      ... .•    d mn _

The proposed method is presented with these following steps.

Step 1. Normalization the decision matrix

In this step, we construct the picture fuzzy decision making matrix. For instance, for all j = 1,2,..., n and i = 1,2,..., m , we calculate

C j

A 1

A 2

c1 jc1 jc1 j 123 c2 jc2 jc2 j c2 mj

3 mj

This step is ignored if matrix D is the given picture fuzzy decision making matrix.

Step 2. Determine the weight of each criteria

We determine the weight A ( j = 1,2,..., n ) of the criteria C ( j = 1,2,..., n ) such that

n

Z A= 1.

j = 1

Step 3. Determine the perfect choice

In this section, we determine the perfect choice. Here, we pay attention to the benefit criteria and cost criteria. Usually, with the perfect choices we can take the picture fuzzy number (1,0,0) for the benefit criteria and (0,0,1) for the cost criteria. Note that (1,0,0) is the largest value of a picture fuzzy linguistic and (0,0,1) is the smallest value of a picture fuzzy linguistic. So that the perfect choice A get the picture fuzzy number A ( j ) at the criteria Cy , in which A ( j ) = (1,0,0) if Cy is the benefit criteria and A ( j ) = (0,0,1) if Cy is the cost criteria, for all j = 1,2,..., n .

Step 4. Calculate the dissimilarity measure of the each alternative to the perfect choice

The dissimilarity measure of the each alternative and the perfect choice Dis( A , A ) .

Step 5. Ranking the alternatives

Now, we can rank the alternatives based on the dissimilarity measure of the each alternative and the perfect choice as follows

A -< Ak iff Dis(A, Ab) < Dis(Ak, Ab)    (11)

• IV.    Our Proposed Method For Evaluating Water Reuse Applications

In this section, we use our method proposed in section 3 for evaluating water reuse application. Data is taken from reference Pan et al [15]. This data is summarized as follows (Table 1, Table 2):

• •     Framework: The proposed has been applied to

the City of Penticton (CoP), British Columbia (BC), Canada. The municipal water is used for residential, commercial, institutional and industrial sectors. The wastewater generated is collected and treated by the an advanced wastewater treatment plant by using biological nutrient removed technology. The treated water is partially reused and the remaining is discharged to a lake.

• •      Specific criteria: C : public acceptability (PA);

C : freshwater saving (FS); C : life cycle cost (LCC);

C : human health risk (HHR); and C : the local governments’ polices (GP).

We consider that C , C , C are the benefit criteria and C , C are the cost criteria.

• •       Alternative water reuses: A : toilet flushing

(TF); A : vegetable watering in gardens (VW); A : flower watering in gardens (FW); A : Agricultural irrigation (ai); A : p ublic parks watering (ppw); A :

Golf course watering (GCW); and A : drinking water (DW).

Now, we present the process of our method for this evaluating water reuse applications.

Step 1 . Normalization the decision matrix .

From the Eq.(10), we obtain the normalization decision matrix (Table 3).

Step 2. Determine the weight of criteria

The weight of criteria C. is ^  =0.2 for all

j = 1,2, ...,5.

Step 3. Determine the perfect choice

The perfect choice is

Ab = ( Ab (1), Ab (2), Ab (3), Ab (4), Ab (5))

where

Ab (1) = Ab (2) = Ab (5) = (1,0,0)

and

Ab (3) = Ab (4) = (0,0,1).

Step 4. Calculate the dissimilarity measure of the each alternative to the perfect choice

Using Eq.(9) with p = 1 , then dissimilarity measure of the each alternative and the perfect choice is calculated by Eq.(8). The results are shown in Table 4.

Step 5. Ranking the alternatives

We use Eq.(11) to rank the alternatives based on the dissimilarity measure of the each alternative and the perfect choice (Table 4).

Table 1. Data of public acceptability and freshwater saving

Alternatives	C : Public acceptability				C : Freshwater saving (ML/year)
	Agree	Neutrality	Disagree		Low	Mid	High
TF	80	9	11		428.8	536	643.2
VW	63.5	13	23.5		2624.8	3281	3937.2
FW	84.5	10	5.5		3192.5	3990.6	4788.8
AI	74.5	10	15.5		3192.5	3990.6	4788.8
PPW	85.5	8	6.5		886.3	1107.9	1329.5
GCW	88.5	7	4.5		361.8	452.3	542.7
DW	24	14	62		3192.5	3990.6	4788.8

Table 2. Data of life cycle cost, human health risk and government’s polices

Alternatives	C 3 : life cycle cost				C4 : human health risk				C 5 : governments’ polices
	Low	Mid	High		Low	Mid	High
TF ( A 1 )	1555358	1944198	2333038		7.10E-12	7.51E-12	8.30E-12		M (0.5,0.4,0.1)
VW( A )	1637219	2046524	2455829		1.83E-11	1.89E-11	2.03E-11		L(0.2,0.5,0.3)
FW( A )	834019	1042524	1251028		1.78E-11	1.84E-11	1.99E-11		H(0.8,0.1,0.05)
AI( A 4 )	146660	183326	219991		9.07E-12	1.00E-11	1.26E-11		M(0.5,0.4,0.1)
PPW( A )	635529	794411	953293		9.34E-12	9.77E-12	1.07E-11		H(0.8,0.1,0.05)
GCW( A 6 )	78219	97774	117328		8.43E-12	8.87E-12	9.83E-12		M(0.5,0.4,0.1)
DW( A )	1197674	1497092	1796511		2.76E-08	4.01E-08	1.00E-07		VL(0.1,0,0.9)

Table 3. Decision matrix

	C 1	C 2	C 3	C 4	C 5
A 1	(0.8,0.09, 0.11)	(0.266667,0.333333, 0.4)	(0.266667,0.333333,0.4)	(0.309908,0.327804,0.362287)	(0.5,0.4,0.1)
A 2	(0.635,0.13,0.235)	(0.266667,0.333333,0.4)	(0.266667,0.333333,0.4)	(0.318261,0.328696,0.353043)	(0.2,0.5,0.3)
A 3	(0.845,0.1,0.055)	(0.266666,0.333331,0.400003)	(0.266667,0.333333,0.4)	(0.317291,0.327986,0.354724)	(0.8,0.1,0.05)
A 4	(0.745,0.1,0.155)	(0.266666,0.333331,0.400003)	(0.266666,0.333334,0.4)	(0.286391,0.315756,0.397853)	(0.5,0.4,0.1)
A 5	(0.855,0.08,0.065)	(0.266661,0.333333,0.400006)	(0.266667,0.333333,0.4)	(0.313318,0.327742,0.35894)	(0.8,0.1,0.05)
A 6	(0.885,0.07,0.045)	(0.266657,0.333358,0.399985)	(0.266667,0.333333,0.399999)	(0.310726,0.326944,0.36233)	(0.5,0.4,0.1)
A 7	(0.24,0.14,0.14)	(0.266666,0.333331,0.400003)	(0.266667,0.333333,0.4)	(0.16458,0.239117,0.596303)	(0.1,0,0.9)

Table 4. Ranking of alternatives with dissimilarity			Now, we give some results when use our method with the difference weight vectors (using Eq.(9) with p = 2 .
Alternatives	DM 1 ( A i , A b )	Rank	For instance, with ^ we consider health-risk’s weight more important than others; with <У2 we ignore the Government policy criteria; and with ( У3 we dismiss the public acceptability criteria. These present in Table 5. Finally, we also recall the results cited in Pan et al. 2018 in Table 6.
TF	0.377652	4
VW	0.451457	6
FW	0.348657	1
AI	0.380642	5
PPW	0.348826	2
GCW	0.363685	3
DW	0.579784	7

Table 5. Ranking of alternatives with dissimilarity and weights

Alternatives	® 1 = (0.1,0.2,0.2,0.4,0.1)		® 2 = (0.25,0.25,0.25,0.25,0)		^ = (0,0.25,0.25,0.25,0.25)
	DM 2 ( A i , A b )	Rank	DM 2 ( A i , A b )	Rank	DM 2 ( A i , A b )	Rank
TF	0.655163	5	0.604568	4	0.636108	4
VW	0.71036	7	0.65549	6	0.716843	6
FW	0.635177	2	0.589318	3	0.596128	2
AI	0.643393	4	0.609762	5	0.624569	3
PPW	0.633343	1	0.589102	2	0.594673	1
GCW	0.645413	3	0.580045	1	0.636155	5
DW	0.704258	6	0.713761	7	0.800051	7

Table 6. Our method’s result and the results cited in Pan et al.[15].

Alt.	Our method with weight ω	Pro-economy	Pro-social	Pro-environment	WRAE with a generalized parameter
TF	4	5	5	5	5
VW	6	6	6	6	6
FW	1	2	1	1	1
AI	5	4	4	3	4
PPW	2	1	2	2	2
GCW	3	3	3	4	3
DW	7	7	7	7	7

• V.    Conclusion

In this paper, we define some new dissimilarity measures of picture fuzzy sets ((Eq.2), Eq.(3) and Eq.(4)). After that, we introduce the MCDM with use the dissimilarity measure of picture fuzzy sets. Finally, we apply the proposed method to evaluate the water reuse applications. We also cite the results were determined in [15] to compare with our method. If the weight vector changes, the results will be also changed. Observe that, our method is easier to understand than the method of Pan [15]. Theory of picture fuzzy set is suitable for data-structured problems such as the problem of water reuse as seen in this paper.