# Optimization-Based TOPSIS Method with Incomplete Weight Information under Nested Probabilistic-Numerical Linguistic Environment

Authors: Yan Deng; Xinxin Wang; Chao Min
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

With the development of the economic and technology, decision-making problems are more and more complex and uncertain. Experts have difficulty in expressing evaluation information because of different research background and insufficient cognition of knowledge structure. Attribute weight information has been often incomplete in decision-making problems. Considering that nested probabilistic-numerical linguistic term sets (NPNLTSs) are flexible to express qualitative and quantitative information, in this paper, we firstly establish an optimization model based on distance measures to obtain the attribute weight. Combined with a classical decision-making method, an optimization-based TOPSIS method with NPNLTSs is proposed to deal with complex decision-making problems. After that, a case study about the river health assessment is given to show the effectiveness and practicability of the proposed method. Finally, some comparative analysis and discussion are provided from three aspects, including the impact for the results without weight optimization, the impact for the results under other uncertain environments, and the impact for the results using other decision-making methods. As a result, the proposed optimization-based TOPSIS method is effective and reliable. The optimization-based TOPSIS method proposed in this paper provides a new way to deal with uncertain and practical problems, which makes a technically sound contribution to the decision-making field.


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## Body

## 1. Introduction

With the rapid development of the economic and society, the uncertainty and complexity of decision-making problems are more and more obvious [1, 2]. In particular, uncertainty is mainly reflected in assessment of alternative, and complexity appears in the information types. At present, decision experts have preferred to express evaluation information by various linguistic models because of practicality and cognitive behavior [3]. Zadeh firstly introduced the fuzzy linguistic approach [3], and some extended linguistic models have been proposed later by scholars [4–7]. For example, hesitant fuzzy linguistic term set (HFLTS) was proposed to express indecisive linguistic information by decision experts [8], and HFLTSs allow to use multiple possible linguistic terms to describe evaluation information but cannot show the weight or possibility of the linguistic term. Later, Pang [9] proposed probabilistic linguistic term set (PLTS) which permits people to use possible linguistic term with the corresponding probability. So far, various linguistic models have been studied by researchers in many aspects, such as operational laws [10–12], distance measures [13, 14], and decision-making methods [15–17]. Considering the type and the structure of evaluation information, nested probabilistic-numerical linguistic term set (NPNLTS) was proposed to help decision experts express qualitative and quantitative information based on PLTS [18]. The characteristics of NPNLTSs are that the linguistic term can be ordinal variable, such as “bad”, “general,” and “good”, and also can be nominal variable, such as “orange”, “jacinth,” and “red”. Moreover, decision experts can express nested evaluation information with NPNLTSs, which is more flexible and accurate to describe uncertain and complex information. And NPNLTSs have been used to deal with multiple attribute group decision-making problems, such as project investment [18] and water resource management [19].In this paper, we study the decision-making method with NPNLTSs under complex and uncertain environment. Under the circumstances, attribute weight has been always incomplete or vague, and it is necessary to obtain the rational attribute weight scientifically because it plays a vital role in aggregation information. Up to now, researchers have proposed various methods to deal with incomplete weight information, such as entropy for objective weights [20], analytic hierarchy process (AHP) [21, 22], minimize deviation [23], and fuzzy for subjective weights [24]. For examples, Jong [25] proposed an extended VIKOR method using incomplete criteria weights, which ranks alternatives using the aggregated scores. Song [26] provided a three-reference-point decision-making method with incomplete weight information considering independent and interactive characteristics. Zhang [27] established two optimization models to minimize the deviation between each decision expert’s evaluation and the group’s collective evaluation on each alternative so that experts can obtain complete weight, due to that the minimize deviation method is effective and useful to get weight information and has been widely applied to decision-making problems [28–31]. Motivated by this method, an optimization model is established to calculate attribute weight under nested probabilistic-numerical linguistic environment.In addition, the TOPSIS method is a classical decision-making method and has been studied in various linguistic environments and fields [32–34] because of its practicability and effectiveness. Specifically, the effects of normalization on the entropy-based TOPSIS method was proposed by Chen to normalize attributes [35]. Aiming at properly dealing with the uncertainty of the decision process, Micale [36] proposed a combined interval-valued ELECTRE TRI and TOPSIS approach for solving the storage location assignment problem. An approach of TOPSIS technique was provided to develop supplier selection with group decision-making under type-2 neutrosophic number [37]. Therefore, in this paper, we mainly study the decision-making problem with incomplete attribute information. An optimization-based TOPSIS method with NPNLTSs is proposed to deal with complex and uncertain problems and apply to a case study considering the river healthy assessment.The contributions in this paper mainly lie in the following aspects: (1) according to incomplete attribute weight information, an optimization model with NPNLTSs is established to obtain scientific attribute weight, which is useful to solve uncertain and practical problems. (2) Combined with the TOPSIS method and NPNLTSs, an optimization-based TOPSIS method is proposed to deal with multiple attribute decision-making problems, and we apply to a case study about river healthy assessment. (3) Some comparative analysis is discussed by simulation experiments from three angles, which are the impact for the results without weight optimization, the impact for the results under other uncertain environments, and the impact for the results using other decision-making methods.To do so, the rest of this paper is organized as follows: in Section2, some concepts and operation of the nested probabilistic-numerical linguistic term sets are presented. Section 3 establishes an optimization model of incomplete weight information based on distance measures. In Section 4, an optimization-based TOPSIS method is proposed to deal with decision-making problems. Section 5 presents a case study about river healthy assessment and makes comparative analysis from three aspects. Section 6 ends the paper with some conclusions.


## 2. Preliminaries

Nested probabilistic-numerical linguistic term set (NPNLTS) is a useful tool to describe uncertain and complex information, especially for qualitative and nested information. In the following, we recall some concepts and operations related to NPNLTS. Firstly, the NPNLTS and the normalized NPNLTS were defined.Definition 1.
(see [18]). Let NPN=OLpILv be a NPNLTS, which consists of an outer-layer probabilistic linguistic term set (OPLTS) OLp and an inner-layer numerical linguistic term set (INLTS) ILv, i.e.,(1)OLp=OLkpkOLk∈OS,pk≥0,k=1,2,…,#OLp,∑k=1#OLppk≤1,(2)ILv=ILklvklILkl∈IS,vkl≥0,k=1,2,…,#OLp,l=1,2,…,#ILv,where OS and IS are an outer-layer linguistic term set (OLTS) and an inner-layer linguistic term set (ILTS), respectively. OLkpk is the k-th outer-layer linguistic term element (OLTE) in the OLTS associated with the probability pk, and #OLp is the number of the linguistic term elements in OLp. ILklvkl is the l-th inner-layer linguistic term element (ILTE) in the ILTS associated with the value vkl under k-th OLTE, and #ILv is the number of the linguistic term elements in ILv.
When the NPNLTSNPN=OLpILv satisfies the following conditions: (1) #OLp equals the number of elements in OS; (2) #ILv equals the number of elements in IS; (3) ∑k=1#OLppk=1; (4) 0≤vkl≤1. Then, the NPNLTS is the normalized NPNLTS (N-NPNLTS).
Score function and comparison laws of the linguistic term sets are two of the basic operations to calculate conveniently for further study. Next, the score function and comparison laws of NPNLTSs were defined based on the definition of NPNLTS.Definition 2.
(see [18]). Let NPN=OLpILv be a NPNLTS, where OLp=OLkpkk=1,…,#OLp and rk be the subscript of OLk in the OLTS, while ILv=ILklvkll=1,…,#ILv and tkl be the subscript of ILkl in ILTS under k-th OLTE. Then, the outer-layer’s score of OLp is(3)OEOLp=sα¯,where α¯=∑k=1#OLprkpk/∑k=1#OLppk, and the inter-layer’s score of ILv is(4)IEILv=v¯klk=1,…,#OLp,l=1,2,…,#ILv,where v¯kl=⌈α¯⌉−α¯×IL⌊α¯⌋vkl+α¯−⌊α¯⌋×IL⌈α¯⌉vkl, ⌈α¯⌉ means the smallest integer greater than α¯, and ⌊α¯⌋ means the greatest integer less than α¯. Then, the whole score of NPN is(5)FNPN=∑l=1#ILvv¯kl#ILv.
The comparison laws of any two NPNLTSs,NPN1 and NPN2, can be presented as follows [18]:(1)
ifFNPN1>FNPN2, then NPN1 is superior to NPN2, denoted by NPN1>NPN2(2)
ifFNPN1<FNPN2, then NPN1 is inferior to NPN2, denoted by NPN1<NPN2(3)
ifFNPN1=FNPN2, then further compared with deviation function. The deviation degree of a NPNLTS is(6)σNPN=∑l=1#ILvvl−FNPN2#ILv1/2.
Therefore, ifσNPN1>σNPN2, then NPN1<NPN2; if σNPN1<σNPN2, then NPN1>NPN2, and if σNPN1=σNPN2, then NPN1 is indifferent to NPN2, denoted by NPN1∼NPN2.
Moreover, two basic operations of INLTSs and OPLTSs in the NPNLTSs are introduced: addition and multiplication, defined as follows.Definition 3 (see [18]).
LetNPN1=OL1kp1kk=1,2,…,#OL1pIL1lv1ll=1,2,…,#IL1v and NPN2=OL2kp2kk=1,2,…,#OL2pIL2lv2ll=1,2,…,#IL2v be two NPNLTSs. For λ1,λ2>0,(7)OL1p⊕OL2p=∪OL1k∈OL1p,OL2k∈OL2pOL1kp1k⊕OL2kp2k,λ1IL1v⊕λ2IL2v=∪IL1l∈IL1v,IL2l∈IL2vλ1IL1lv1l⊕λ2IL2lv2l,OL1p⊗OL2p=∪OL1k∈OL1p,OL2k∈OL2pOL1kp1k⊗OL2kp2k,λ1IL1v⊗λ2IL2v=∪IL1l∈IL1v,IL2l∈IL2vIL1lv1lλ1⊗IL2lv2lλ2,where OL1k and OL2k are the k-th linguistic terms and p1k and p2k are the probabilities of the k-th linguistic terms in OL1k and OL2k, respectively. IL1l and IL2l are the l-th linguistic terms and v1l and v2l are the values of the l-th linguistic terms in IL1v and IL2v, respectively.


## 3. Optimization Model of Incomplete Weight Information

In this section, we introduce incomplete weight information about attributes in decision-making problems and further establish an optimization model to calculate the weights.As for a multiple attributes group decision-making problem under the nested probabilistic-numerical linguistic environment, letA=A1,A2,…,Am be a set of alternatives, C=C1,C2,…,Cn be a set of attributes, and ω=ω1,ω2,…,ωnT be the corresponding attribute vector, which satisfies ∑j=1nωj=1 and ωj≥0, j=1,2,…,n. E=E1,E2,…,Ek is a set of experts, and w=w1,w2,…,wkT is the associated weight vectors over experts, with wq≥0 and ∑q=1kwq=1. Let OS=s0,s1,…,sτ be the OLTS, IS=n0,n1,…,nς be the ILTS, and NPNq=NPNijqm×nq=1,2,…,k be the decision matrix given by the expert Eq∈E, where NPNijq represents the evaluation information for alternative Ai∈A with respect to the attribute Cj∈C:(8)NPNq=NPN11qNPN12q⋯NPN1nqNPN21qNPN22q⋯NPN2nq⋮⋮⋱⋮NPNm1qNPNm2q⋯NPNmnq.Since the complex and uncertain environment, experts give the incomplete weight information for attributes. Up to now, the incomplete weight has been mainly divided into six cases [38]: (1) ωi≥ωj; (2) ωi−ωj≥αi; (3) ωi≥βiωj; (4) γi+εi≥ωi≥γi; (5) θi+εiωj≥θiωj or θi≤ωi/ωj≥θi+εi,ωj≠0; (6) ωi−ωj≥ωk−ωl,j≠k≠l, where αi,βi,γi,θi, and εi are nonnegative constants.As for a finite number of alternatives, the essence of the multiattribute decision-making is to compare and rank the comprehensive attribute values of alternatives. Considering the attribute values of alternatives with respect to attributeCj, if the difference is smaller, then the effect of the attribute Cj on the decision results is smaller. On the contrary, if the difference is larger, it shows that the attribute Cj plays an important role during decision-making process. From the perspective of alternatives’ ranking, the attribute of the alternative with greater deviation should be given greater weight. In particular, if the attribute values of alternatives are the same, then the attribute Cj do not work on the alternatives’ ranking and then let the attribute weight be 0. As we can see, this way can evaluate the weight of attributes objectively.For any attributeCj in the decision matrix NPNq=NPNijqm×nq=1,2,…,k, distance measure is used to represent the deviation Diω between alternative Ai and alternative Ak. According to the generalized weighted distance measure with nested probabilistic-numerical linguistic information [19], the deviation Diω is defined as follows.Definition 4.
LetNPN=NPNijm×n be the decision matrix with nested probabilistic-numerical linguistic information. ω=ω1,ω2,…,ωnT is the weight vector of attributes, where 0≤ωj≤1 and ∑j=1nωj=1, and the deviation Diω between alternative Ai and alternative Ak based on the generalized weighted distance measure is defined as(9)Diω=∑k=1m∑j=1ndNPNij,NPNkj=∑k=1m∑j=1nωj#OL×#IL∑k=1#OL∑l=1#ILOLijk−OLkjk/τ+1+pijk−pkjk+ILijl−ILkjl/ς+1+vijl−vkjl4λ1/λ,where λ>0, #OL is the number of the linguistic term elements in OLTS, and #IL is the number of the linguistic term elements in ILTS. In particular, when λ=1 and λ=2, the generalized weighted distance measure reduces to Hamming weighted distance measure and Euclid weighted distance measure, respectively.
For all attributesCjj=1,2,…,n, Dω denotes the sum of deviation between alternative Ai and other alternatives. In order to determine the weight vector ω, we need to maximize the Dω. Therefore, the objective function is as follows:(10)maxDω=∑i=1mDiω=∑i=1m∑k=1m∑j=1nωj#OL×#IL∑k=1#OL∑l=1#ILOLijk−OLkjk/τ+1+pijk−pkjk+ILijl−ILkjl/ς+1+vijl−vkjl4λ1/λ.
And the optimization model is established to obtain the weight vector of attributes:
Model I:(11)maxDωs.t.∑j=1nωj=1,ωj≥0,j=1,2,…,n.


## 4. Optimization-Based TOPSIS Method

In this section, an optimization-based TOPSIS method with the nested probabilistic-numerical linguistic information is proposed, and the corresponding algorithm is presented.


### 4.1. The Extended TOPSIS Method

TOPSIS method is a classical method, and it has been widely applied to many fields. Considering incomplete weight information of attributes, we propose an extended TOPSIS method with the nested probabilistic-numerical linguistic information. Firstly, we define some concepts related to the extended TOPSIS method.Definition 5.
LetNPN=NPNijm×n=OLijpILvm×n be the decision matrix with the nested probabilistic-numerical linguistic information. For the alternative Aii=1,2,…,m, NPNi=NPNi1,NPNi2,…,NPNin is the attribute vector. NPN+=NPN1+,NPN2+,…,NPNn+ is the positive ideal solution (PIS) of the alternative, where(12)NPNj+=OLjkIL+k=1,2,…,#OLij,OLjkIL+=skmaxipijkIL,j=1,2,…,n,NPN−=NPN1−,NPN2−,…,NPNn−is the negative ideal solution (NIS) of the alternative, where(13)NPNj−=OLjkIL−k=1,2,…,#OLij,OLjkIL−=skminipijkIL,j=1,2,…,n.
According to the decision matrixNPN=NPNijm×n, the PIS and the NIS are(14)NPN+=NPN1+,NPN2+,…,NPNn+,NPN−=NPN1−,NPN2−,…,NPNn−,respectively. The deviation between each alternative and the PIS is defined as follows:(15)dAi,NPN+=ωjdNPNij,NPNj+=ωj1#OLij∑k=1#OLijOLijk−OLij+/τ+1+pijk−pij+221/2,and the deviation between each alternative and the NIS is(16)dAi,NPN−=ωjdNPNij,NPNj−=ωj1#OLij∑k=1#OLijOLijk−OLij−/τ+1+pijk−pij−221/2.
The smaller the deviationdAi,NPN+, the better the alternative Ai, and the larger the deviation dAi,NPN−, the better the alternative Ai. Let(17)dminAi,NPN+=min1≤i≤mdAi,NPN+be the smallest deviation between the alternative Ai and the PIS,(18)dmaxAi,NPN−=max1≤i≤mdAi,NPN−,be the largest deviation between the alternative Ai and the NIS. Motivated by closeness coefficient [39], the inner weight of the alternative is(19)εiAi=dAi,NPN−/dmaxAi,NPN−+dAi,NPN+/dminAi,NPN+∑i=1mdAi,NPN−/dmaxAi,NPN−+dAi,NPN+/dminAi,NPN+.
It is noted that the inner weight is a vector, that is,εi=εi0,εi1,…,εiτT. And the inner-layer score IZi is expressed as(20)IZi=εi0ILi1⊕εi1ILi2⊕⋯⊕εiτILin=∪ILi1l∈ILi1vεi0ILi1lvi1l⊕∪ILi2l∈ILi2vεi1ILi2lvi2l⊕⋯⊕∪ILinl∈ILinvεiτILinlvinl,where i=1,2,…,m.
According to Definition2, the whole score FNPNi of each alternative Ai can be obtained with the inner weight εi=εi0,εi1,…,εiτT. Hence, we can rank the alternatives and select the best one. In particular, the comparison rule between the alternative Ai and the alternative Ari,r=1,2,…,m;i≠r is that(1)
IfFNPNi>FNPNr, then Ai>Ar(2)
IfFNPNi<FNPNr, then Ai<Ar(3)
IfFNPNi=FNPNr, further compare σNPNi and σNPNr:(a)
σNPNi>σNPNr, then Ai<Ar(b)
σNPNi<σNPNr, then Ai>Ar(c)
σNPNi=σNPNr, then Ai∼Ar


### 4.2. The Procedure Based on Incomplete Weight Information

In order to deal with a multiattribute group decision-making problem under the nested probabilistic-numerical linguistic environment, the extended TOPSIS method with incomplete weight information is divided into four parts: (1) preparation part. Analyze the decision-making problem, and determine a set of alternatives, attributes, and experts. (2) Optimization part. Optimize attribute weight with incomplete information by Model I. (3) Evaluation part. Construct the decision matrix with NPNLTSs by experts and calculate the inner weight. (4) Decision part. Calculate the whole scores of alternatives and select the best one. In the following, we propose the algorithm of the extended TOPSIS method with incomplete weight information (Algorithm1).Algorithm 1:The extended TOPSIS method with incomplete weight information.
Step 1. Determine a set of alternativesA=A1,A2,…,Am and a set of attributes C=C1,C2,…,Cn by experts according to the decision-making problem. Go to Step 2.Step 2. Provide weight information about attributes by experts, and obtain the optimized weight vectorω=ω1,ω2,…,ωnT by Model I. Go to Step 3.Step 3. Give the OLTSOS=sαα=0,1,…,τ and the ILTS IS=nββ=0,1,…,ς by experts. Go to Step 4.Step 4. Construct the OPLTSOLijp=OLijkpijkk=1,2,…,#OLijp over the alternatives Aii=1,2,…,m with respect to the attributes Cjj=1,2,…,n, where the corresponding probability pijk>0. Go to Step 5.Step 5. Construct the INLTSILαβv=ILαβlvαβll=0,1,…,ς based on OS=sαα=0,1,…,τ, where the corresponding value vαβl>0. Go to Step 6.Step 6. Get the PISNPN+ and the NIS NPN− of the alternatives by Definition 5. Go to Step 7.Step 7. Calculate the deviationdAi,NPN+ between the alternative and PIS by equation (15), and the deviation dAi,NPN− between the alternative and NIS by equation (16). Go to Step 8.Step 8. Get the smallest deviationdminAi,NPN+ and the largest deviation dmaxAi,NPN−. Go to Step 9.Step 9. Calculate the inner weightεiAi of the alternative Aii=1,2,…,m by equation (19). Go to Step 10.Step 10. Calculate the inner-layer scoreIZi of the alternative Aii=1,2,…,m by equation (20). Go to Step 11.Step 11. Calculate the whole scoreFNPNi of the alternative Aii=1,2,…,m by Definition 2. Go to Step 12.Step 12. Rank the alternativesAii=1,2,…,m by comparison rule and select the best alternative. Go to Step 13.Step 13. End.In order to better understand the procedure, Figure1 shows the flowchart of the algorithm in decision-making with NPNLTSs.Figure 1
The flowchart of the optimization-based TOPSIS method algorithm with NPNLTSs.


## 4.1. The Extended TOPSIS Method

TOPSIS method is a classical method, and it has been widely applied to many fields. Considering incomplete weight information of attributes, we propose an extended TOPSIS method with the nested probabilistic-numerical linguistic information. Firstly, we define some concepts related to the extended TOPSIS method.Definition 5.
LetNPN=NPNijm×n=OLijpILvm×n be the decision matrix with the nested probabilistic-numerical linguistic information. For the alternative Aii=1,2,…,m, NPNi=NPNi1,NPNi2,…,NPNin is the attribute vector. NPN+=NPN1+,NPN2+,…,NPNn+ is the positive ideal solution (PIS) of the alternative, where(12)NPNj+=OLjkIL+k=1,2,…,#OLij,OLjkIL+=skmaxipijkIL,j=1,2,…,n,NPN−=NPN1−,NPN2−,…,NPNn−is the negative ideal solution (NIS) of the alternative, where(13)NPNj−=OLjkIL−k=1,2,…,#OLij,OLjkIL−=skminipijkIL,j=1,2,…,n.
According to the decision matrixNPN=NPNijm×n, the PIS and the NIS are(14)NPN+=NPN1+,NPN2+,…,NPNn+,NPN−=NPN1−,NPN2−,…,NPNn−,respectively. The deviation between each alternative and the PIS is defined as follows:(15)dAi,NPN+=ωjdNPNij,NPNj+=ωj1#OLij∑k=1#OLijOLijk−OLij+/τ+1+pijk−pij+221/2,and the deviation between each alternative and the NIS is(16)dAi,NPN−=ωjdNPNij,NPNj−=ωj1#OLij∑k=1#OLijOLijk−OLij−/τ+1+pijk−pij−221/2.
The smaller the deviationdAi,NPN+, the better the alternative Ai, and the larger the deviation dAi,NPN−, the better the alternative Ai. Let(17)dminAi,NPN+=min1≤i≤mdAi,NPN+be the smallest deviation between the alternative Ai and the PIS,(18)dmaxAi,NPN−=max1≤i≤mdAi,NPN−,be the largest deviation between the alternative Ai and the NIS. Motivated by closeness coefficient [39], the inner weight of the alternative is(19)εiAi=dAi,NPN−/dmaxAi,NPN−+dAi,NPN+/dminAi,NPN+∑i=1mdAi,NPN−/dmaxAi,NPN−+dAi,NPN+/dminAi,NPN+.
It is noted that the inner weight is a vector, that is,εi=εi0,εi1,…,εiτT. And the inner-layer score IZi is expressed as(20)IZi=εi0ILi1⊕εi1ILi2⊕⋯⊕εiτILin=∪ILi1l∈ILi1vεi0ILi1lvi1l⊕∪ILi2l∈ILi2vεi1ILi2lvi2l⊕⋯⊕∪ILinl∈ILinvεiτILinlvinl,where i=1,2,…,m.
According to Definition2, the whole score FNPNi of each alternative Ai can be obtained with the inner weight εi=εi0,εi1,…,εiτT. Hence, we can rank the alternatives and select the best one. In particular, the comparison rule between the alternative Ai and the alternative Ari,r=1,2,…,m;i≠r is that(1)
IfFNPNi>FNPNr, then Ai>Ar(2)
IfFNPNi<FNPNr, then Ai<Ar(3)
IfFNPNi=FNPNr, further compare σNPNi and σNPNr:(a)
σNPNi>σNPNr, then Ai<Ar(b)
σNPNi<σNPNr, then Ai>Ar(c)
σNPNi=σNPNr, then Ai∼Ar


## 4.2. The Procedure Based on Incomplete Weight Information

In order to deal with a multiattribute group decision-making problem under the nested probabilistic-numerical linguistic environment, the extended TOPSIS method with incomplete weight information is divided into four parts: (1) preparation part. Analyze the decision-making problem, and determine a set of alternatives, attributes, and experts. (2) Optimization part. Optimize attribute weight with incomplete information by Model I. (3) Evaluation part. Construct the decision matrix with NPNLTSs by experts and calculate the inner weight. (4) Decision part. Calculate the whole scores of alternatives and select the best one. In the following, we propose the algorithm of the extended TOPSIS method with incomplete weight information (Algorithm1).Algorithm 1:The extended TOPSIS method with incomplete weight information.
Step 1. Determine a set of alternativesA=A1,A2,…,Am and a set of attributes C=C1,C2,…,Cn by experts according to the decision-making problem. Go to Step 2.Step 2. Provide weight information about attributes by experts, and obtain the optimized weight vectorω=ω1,ω2,…,ωnT by Model I. Go to Step 3.Step 3. Give the OLTSOS=sαα=0,1,…,τ and the ILTS IS=nββ=0,1,…,ς by experts. Go to Step 4.Step 4. Construct the OPLTSOLijp=OLijkpijkk=1,2,…,#OLijp over the alternatives Aii=1,2,…,m with respect to the attributes Cjj=1,2,…,n, where the corresponding probability pijk>0. Go to Step 5.Step 5. Construct the INLTSILαβv=ILαβlvαβll=0,1,…,ς based on OS=sαα=0,1,…,τ, where the corresponding value vαβl>0. Go to Step 6.Step 6. Get the PISNPN+ and the NIS NPN− of the alternatives by Definition 5. Go to Step 7.Step 7. Calculate the deviationdAi,NPN+ between the alternative and PIS by equation (15), and the deviation dAi,NPN− between the alternative and NIS by equation (16). Go to Step 8.Step 8. Get the smallest deviationdminAi,NPN+ and the largest deviation dmaxAi,NPN−. Go to Step 9.Step 9. Calculate the inner weightεiAi of the alternative Aii=1,2,…,m by equation (19). Go to Step 10.Step 10. Calculate the inner-layer scoreIZi of the alternative Aii=1,2,…,m by equation (20). Go to Step 11.Step 11. Calculate the whole scoreFNPNi of the alternative Aii=1,2,…,m by Definition 2. Go to Step 12.Step 12. Rank the alternativesAii=1,2,…,m by comparison rule and select the best alternative. Go to Step 13.Step 13. End.In order to better understand the procedure, Figure1 shows the flowchart of the algorithm in decision-making with NPNLTSs.Figure 1
The flowchart of the optimization-based TOPSIS method algorithm with NPNLTSs.


## 5. A Case Study

In this Section, a multiple attribute group decision-making problem is considered about the river health assessment. The optimization-based TOPSIS method is used to deal with the problem, and we further make some comparative analysis considering three factors.


### 5.1. Problem Description

River health assessment is based on the connotation analysis, in view of the river natural, ecological, and social service functions. The connotation of the river health is that under the utilization and protection coordination of human, natural rivers, ecological function, and social service function relatively balanced the state of play, the river can realize the normal water, basic material, and energy circulation and good function, including to maintain a certain level of ecological environment and social service function and meet the demand of the sustainable development of human society, eventually form human balance development and protection of rivers of virtuous circle. According to the basic characteristics of the river and the individual characteristics, river health assessment sets up the river health evaluation system by the commonness and individuality indexes and puts forward the overall evaluation method from river to river.River health is supposed to include the natural health of the river and the good ecological environment and social service. However, in the current social and economic background in China, several river basins are densely populated and highly developed, and it is difficult to realize the natural, ecological, and social service functions of rivers. Therefore, the river health should provide relatively good ecological environment and the social service function, and meet the needs of sustainable development of human society, the corresponding period, namely, in keeping the river natural, ecological, and social service function of a state of equilibrium. At present, some factors are popular to be considered for evaluating the river health as follows:Monitoring status assessment of the river. The scope of monitoring is very broad, including regular surface and groundwater monitoring, process monitoring, and emergency incident monitoring. Monitoring status assessment can provide data for environmental management and provide a basis for water quality assessment of the river.The evaluation of sustainable use of water resources. Sustainable utilization of water resources is the principle to ensure the sustainable development of human society, economy, and living environment. Life support systems and ecosystems on earth must not be compromised in the development and utilization of water resources.The impact assessment of the hydraulic engineering. It refers to the analysis, prediction, and evaluation of the environmental quality changes caused by the construction of water conservancy projects. The scope depends on the scale, characteristics, and geographical location of the projects.The purpose of studying and maintaining river health is to restore the river's natural function and make it play its social function in a balanced way, so as to maintain the sustainable utilization of river's social function and guarantee the sustainable development of human economy and society. In the following, we will make the river health assessment for four rivers, which are the Yangtze River, the Yellow River, the Pearl River, and the Huai River.


### 5.2. Solve the Problem

According to Section5.1, four evaluation objects are put forward, and denote as X=x1,x2,x3,x4, where x1,x2,x3,x4 are the Yangtze River, the Yellow River, the Pearl River, and the Huai River, respectively. At the same time, we consider three attributes C=c1,c2,c3, where c1 is monitoring status assessment of the river, c2 is the evaluation of sustainable use of water resources, and c3 is the impact assessment of the hydraulic engineering. In order to evaluate the river health comprehensively, the OLTS and the ILTS are given as follows:(21)OS=s0=very bad,s1=bad,s2=medium,s3=good,s4=very good,IS=n0=river hydrological condition,n1=water environment condition,n2=hydrophytic habitat,n3=water resources development and utilization.Experts express their evaluation information with NPNLTSs. Specifically, they evaluate objects firstly with respect to the OLTS and assess each outer linguistic term in terms of the ILTS. Therefore, a nested probabilistic-numerical linguistic decision matrix is constructed with NPNLTSs, shown as Table1.Table 1
The nested probabilistic-numerical linguistic decision matrix.
c1c2c3x1s00.1n00.2,n10.4,n20.4,s10.1n10.6,n20.4,s20.2n20.2,n30.3,s30.1n30.8s00.2n20.4,s10.5n10.2,n20.2,n20.1,s20.1n20.4,n30.6,s30.2n10.3,n20.2,n30.5s00.2n00.2,n10.4,n20.2,n20.2,s10.2n20.6,n30.4,s20.2n00.4,n10.3,n20.3,s30.2n20.2,n30.8x2s00.5n20.4,s10.2n10.2,n20.3,s20.2n10.2,n20.2,n30.1,s30.1n20.2,n30.8s00.2n00.4,n10.4,s10.3n00.2,n10.2,n20.4,s20.3n10.3,n20.2,n30.3,s30.2n00.1,n10.3,n20.3,n30.3s00.3n10.6,n20.4,s10.2n10.2,n20.3,s20.3n20.1,n30.4,s30.2n20.2,n30.8x3s00.1n00.3,n10.5,s10.2n00.2,n10.4,n20.4,s20.3n20.4,n30.6,s30.4n20.5,n30.5s00.4n00.1,n10.2,n20.2,n30.5,s10.4n10.3,n20.2,s20.1n20.2,n30.8,s30.1n20.4,n30.6s00.2n00.1,n10.3,n20.6,s10.1n10.3,n20.4,n30.3,s20.1n20.2,n30.3,s30.1n20.7,n30.3x4s00.1n20.4,s10.1n10.5,n20.5,s20.1n20.2,n30.3,s30.2n20.4,n30.6s00.1n00.1,n10.4,n20.5,s10.2n10.5,n20.5,s20.1n10.2,n20.2,n30.1,s30.1n10.3,n20.7s00.2n00.1,n10.2,n20.2,s10.2n10.5,n20.5,s20.3n20.2,n30.3,s30.3n10.2,n20.4,n30.4After normalization of attributes, the normalized nested probabilistic-numerical linguistic decision matrix can be shown in Table2.Due to the complexity and uncertainty of the assessment environment, experts give the incompletely weight information about the attributes as follows:(22)0.2≤ω1≤0.4,0.3≤ω2≤0.5,0.2≤ω3≤0.4,0.1ω2≤ω3,ωj≥0,j=1,2,3,∑j=13ωj=1.According to Model I, the optimization model is established to calculate the weight vector of attributes, which is shown as follows:(23)maxDω=∑i=14Diω=∑j=13∑i=14∑k=14ωj4×5∑k=14∑l=15OLijk−OLkjk/4+1+pijk−pkjk+ILijl−ILkjl/5+1+vijl−vkjl4λ1/λs.t.0.2≤ω1≤0.4,0.3≤ω2≤0.5,0.2≤ω3≤0.4,0.1ω2≤ω3,∑j=13ωj=1,ωj≥0,j=1,2,3.Suppose that the parameterλ=1, and the model is simplified as(24)maxDω=0.45ω1+0.5563ω2+0.4525ω3s.t.0.2≤ω1≤0.4,0.3≤ω2≤0.5,0.2≤ω3≤0.4,0.1ω2≤ω3,∑j=13ωj=1,ωj≥0,j=1,2,3.The weight vector can be obtained by Lingo 11, and the result isω=0.2,0.5,0.3T.Next, the proposed extended TOPSIS method is used to evaluate the river health, and the following steps are as follows:Table 2
The normalized nested probabilistic-numerical linguistic decision matrix.
c1c2c3x1s00.2n00.2,n10.4,n20.4,s10.2n10.6,n20.4,s20.4n20.4,n30.6,s30.2n31s00.2n21,s10.5n10.4,n20.4,n30.2,s20.1n20.4,n30.6,s30.2n10.3,n20.2,n30.5s00.25n00.2,n10.4,n20.2,n30.2,s10.25n20.6,n30.4,s20.25n00.4,n10.3,n20.3,s30.25n20.2,n30.8x2s00.5n21,s10.2n10.4,n20.6,s20.2n10.4,n20.4,n30.2,s30.1n20.2,n30.8s00.2n00.5,n10.5,s10.3n00.25,n10.25,n20.5,s20.3n10.375,n20.25,n30.375,s30.2n00.1,n10.3,n20.3,n30.3s00.3n10.6,n20.4,s10.2n10.4,n20.6,s20.3n20.2,n30.8,s30.2n20.2,n30.8x3s00.1n00.375,n10.625,s10.2n00.2,n10.4,n20.4,s20.3n20.4,n30.6,s30.4n20.5,n30.5s00.4n00.1,n10.2,n20.2,n30.5,s10.4n10.6,n20.4,s20.1n20.2,n30.8,s30.1n20.4,n30.6s00.4n00.1,n10.3,n20.6,s10.2n10.3,n20.4,n30.3,s20.2n20.4,n30.6,s30.2n20.7,n30.3x4s00.2n21,s10.2n10.5,n20.5,s20.2n20.4,n30.6,s30.4n20.4,n30.6s00.2n00.1,n10.4,n20.5,s10.4n10.5,n20.5,s20.2n10.4,n20.4,n30.2,s30.2n10.3,n20.7s00.2n00.2,n10.4,n20.4,s10.2n10.5,n20.5,s20.3n20.4,n30.6,s30.3n10.2,n20.4,n30.4Table 3
The positive ideal solutionNPN+ and the negative ideal solution NPN−.
c1c2c3NPN+s00.1n00.375,n10.625,s10.2n00.2,n10.4,n20.4,s20.3n20.4,n30.6,s30.4n20.5,n30.5s00.2n00.5,n10.5,s10.3n00.25,n10.25,n20.5,s20.3n10.375,n20.25,n30.375,s30.2n00.1,n10.3,n20.3,n30.3s00.2n00.2,n10.4,n20.4,s10.2n10.5,n20.5,s20.3n20.4,n30.6,s30.3n10.2,n20.4,n30.4NPN−s00.5n21,s10.2n10.4,n20.6,s20.2n10.4,n20.4,n30.2,s30.1n20.2,n30.8s00.4n00.1,n10.2,n20.2,n30.5,s10.4n10.6,n20.4,s20.1n20.2,n30.8,s30.1n20.4,n30.6s00.4n00.1,n10.3,n20.6,s10.2n10.3,n20.4,n30.3,s20.2n20.4,n30.6,s30.2n20.7,n30.3Step 1. Considering that attributes are all benefit-type, the PISNPN+ and the NIS NPN− of the objects are listed in Table 3 according to Table 2.Step 2. According to equations (15) and (16), the deviation dxi,NPN+ and dxi,NPN− can be calculated as follows:(25)dxi,NPN+=0.0785,0.0452,0.0800,0.0516,dxi,NPN−=0.1224,0.0257,0.0538,0.0727.Step 3. Getdminxi,NPN+=0.0452 and dmaxxi,NPN−=0.1224.Step 4. According to equation (19), the inner weight εixi of the object can be calculated as follows:(26)εixi=0.3430,0.1556,0.2833,0.2182T.Step 5. According to equation (20), the inner-layer score IZi of the object can be calculated as follows:(27)IZ1=0.12530.25050.34940.0567000.26800.36940.144400.11330.08500.28440.2991000.04670.08780.64740,IZ2=0.07780.24770.4563000.03890.28940.45350000.19550.23270.353600.01560.04670.17190.54770,IZ3=0.17250.33050.20110.077800.06860.31550.31270.08500000.28160.50020000.43200.34980,IZ4=0.07220.17550.53410000.39090.39090000.06220.31270.4069000.10330.35940.31910.Step 6. According to Definition2, the whole score FNPNi of the object are(28)FNPNi=0.8071,0.7602,0.7227,0.7653.Step 7. Rank the objects by comparison rule, that is,x1>x4>x2>x3, and the best object is the Yangtze River.


### 5.3. Comparative Analysis and Discussion

In order to verify the superiority of the proposed method, we make some comparative analysis based on simulation experiments by MATLAB from three aspects: (1) the impact for the results without weight optimization; (2) the impact for the results under other uncertain environments; (3) the impact for the results using other decision-making methods.


#### 5.3.1. The Impact without Weight Optimization

Suppose that the attribute weights satisfy several cases, such as uniform distribution, increase progressively, decrease progressively, and irregular situation. With different attribute weights, the whole scores of the objects and the rankings are shown in Table4.Table 4
The whole scores of the objects and the rankings with different attribute weights.
ωFx1Fx2Fx3Fx4Rankingsω=1/3,1/3,1/3T0.74440.70080.63880.6921x1>x2>x4>x3ω=0.1,0.3,0.6T0.78330.73090.68190.7318x1>x4>x2>x3ω=0.6,0.3,0.1T0.69630.66020.58030.6396x1>x4>x2>x3ω=0.4,0.5,0.1T0.76560.72480.67380.7207x1>x2>x4>x3ω=0.2,0.5,0.3T0.80710.76020.72270.7653x1>x4>x2>x3As we can see, different attribute weights have effect on the whole scores of the objects, and the rankings are also a little different. Although the best object and the worst object arex1 and x3, respectively, the rank of x2 and x4 are different because of the various attribute weights. Therefore, it can be concluded that when the numbers of objects and attributes are large, it is necessary to make full use of information to optimize the weight. Because various attribute weights may lead to different ranking results, it is also one of the most important factors to obtain a rational result.In the following, we further analyze the relationship between the attribute weights and the whole scores of objects. We generate 1000 sets of attribute weights randomly based on MATLAB and apply to the same case, and the whole scores of each object can be shown in Figure2.Figure 2
The whole scores of each object with 1000 sets of attribute weights: (a) two-dimensional diagram; (b) three-dimensional diagram.
(a)(b)In general, the range of the whole scores is from 0.5 to 0.95. In Figure2(a), the whole scores of x1 are almost the highest, while the whole scores of x3 are almost the lowest. And the whole scores of x2 and x4 are larger and smaller alternatively. In Figure 2(b), we can see the trend clearly with various attribute weights. Specifically, the maximum values of the whole scores of four objectives are 0.9384, 0.8950, 0.8942, and 0.9167, respectively. The minimum values of the whole scores of four objectives are 0.6623, 0.6234, 0.5064, and 0.5857, respectively. Therefore, the attribute weights can affect the whole scores about 20%, and it is meaningful to determine the weights rationally and scientifically.


#### 5.3.2. The Impact under Other Uncertain Environments

Next, we study the case using different linguistic evaluation methods. In Section5.2, experts use the NPNLTSs to express the evaluation information considering outer and inner linguistic information. Then, hesitant fuzzy linguistic term sets (HFLTSs) [8] and probabilistic linguistic term sets (PLTSs) [9] are used to assess evaluation information of the objects with respect to the attributes, respectively. Table 5 and 6 list the evaluation information with HFLTSs and PLTSs, respectively.Table 5
The evaluation information with HFLTSs.
c1c2c3x1s2,s3s1,s2,s3s0,s1,s2,s3x2s2s1,s2,s3s2,s3x3s2,s3s0,s1,s2,s3s0,s1,s2x4s2,s3s1,s2s1,s2,s3Table 6
The evaluation information with PLTSs.
c1c2c3x1s00.2,s10.2,s20.4,s30.2s00.2,s10.5,s20.1,s30.2s00.25,s10.25,s20.25,s30.25x2s00.5,s10.2,s20.2,s30.1s00.2,s10.3,s20.3,s30.2s00.3,s10.2,s20.3,s30.2x3s00.1,s10.2,s20.3,s30.4s00.4,s10.4,s20.1,s30.1s00.4,s10.2,s20.2,s30.2x4s00.2,s10.2,s20.2,s30.4s00.2,s10.4,s20.2,s30.2s00.2,s10.2,s20.3,s30.3Similarly, we use TOPSIS methods with HFLTSs and PLTSs to deal with the case, and the rankings are listed in Table7.Table 7
Rankings with different linguistic models.
MethodsRankingsThe TOPSIS method with HFLTSs [40]x4>x1>x3>x2The TOPSIS method with PLTSs [9]x1>x2>x4>x3The proposed methodx1>x4>x2>x3The ranking results show thatx1 is the best object by TOPSIS methods with PLTSs and NPNLTSs, while x4 is the best object by TOPSIS methods with HFLTSs, and the rankings are different by TOPSIS methods with three various linguistic models. The reason may be that HFLTSs do not consider the weight of linguistic terms, and PLTSs cannot better reflect information comprehensively according to Tables 5 and 6. Therefore, the results by the decision method with NPNLTSs are more reasonable and reliable.


#### 5.3.3. The Impact Using Other Decision-Making Methods

Finally, we make analysis to compare the proposed method with other decision-making methods with NPNLTSs, which are aggregation method [18], VIKOR method [41], and ELECTRE method [42]. Suppose that the attribute weights are ω=0.2,0.5,0.3T, four methods above are used to deal with the same case, and the ranking results are listed in Table 8.Table 8
Rankings by four decision-making methods.
MethodsRankingsThe aggregation method [18]x1>x4>x3>x2The VIKOR method [41]x1>x4>x2>x3The ELECTRE method [42]x1>x2>x4>x3The proposed methodx1>x4>x2>x3As we can see from Table8, ranking results by four decision-making methods are a little different. Specifically, all the best objects are x1, and the rankings are the same by the proposed method and the VIKOR method. To some extent, the proposed TOPSIS method is reliable.In order to further study the effectiveness of the proposed method, we make simulation experiments to analyze the average operation time (AOT) by four decision-making methods. Letm and k be the number of objects and attributes, respectively. When m and k are from 3 to 12 after 1000 simulation times, the AOTs by four decision-making methods are shown in Figure 3.Figure 3
The average operation time by four decision-making methods: (a)m is from 3 to 12 with k=3; (b) k is from 3 to 12 with m=3.
(a)(b)Whenm and k are from 3 to 12, AOTs with four decision-making methods are all gradually increasing, and there are apparent differences by various methods. In Figure 3(a), the aggregation method takes the minimum AOT, and the ELECTRE method takes the maximum AOT. The AOT of the proposed method is little more than the aggregation method. In Figure 3(b), the minimum AOT also belongs to the aggregation method, while the maximum AOT belongs to the ELECTRE method. The proposed method also takes more time than the aggregation method. In addition, increasing the number of objects would take more time than increasing the number of attributes. Therefore, the proposed method is effective without taking too much time, and the results by the proposed method are also reliable.


## 5.1. Problem Description

River health assessment is based on the connotation analysis, in view of the river natural, ecological, and social service functions. The connotation of the river health is that under the utilization and protection coordination of human, natural rivers, ecological function, and social service function relatively balanced the state of play, the river can realize the normal water, basic material, and energy circulation and good function, including to maintain a certain level of ecological environment and social service function and meet the demand of the sustainable development of human society, eventually form human balance development and protection of rivers of virtuous circle. According to the basic characteristics of the river and the individual characteristics, river health assessment sets up the river health evaluation system by the commonness and individuality indexes and puts forward the overall evaluation method from river to river.River health is supposed to include the natural health of the river and the good ecological environment and social service. However, in the current social and economic background in China, several river basins are densely populated and highly developed, and it is difficult to realize the natural, ecological, and social service functions of rivers. Therefore, the river health should provide relatively good ecological environment and the social service function, and meet the needs of sustainable development of human society, the corresponding period, namely, in keeping the river natural, ecological, and social service function of a state of equilibrium. At present, some factors are popular to be considered for evaluating the river health as follows:Monitoring status assessment of the river. The scope of monitoring is very broad, including regular surface and groundwater monitoring, process monitoring, and emergency incident monitoring. Monitoring status assessment can provide data for environmental management and provide a basis for water quality assessment of the river.The evaluation of sustainable use of water resources. Sustainable utilization of water resources is the principle to ensure the sustainable development of human society, economy, and living environment. Life support systems and ecosystems on earth must not be compromised in the development and utilization of water resources.The impact assessment of the hydraulic engineering. It refers to the analysis, prediction, and evaluation of the environmental quality changes caused by the construction of water conservancy projects. The scope depends on the scale, characteristics, and geographical location of the projects.The purpose of studying and maintaining river health is to restore the river's natural function and make it play its social function in a balanced way, so as to maintain the sustainable utilization of river's social function and guarantee the sustainable development of human economy and society. In the following, we will make the river health assessment for four rivers, which are the Yangtze River, the Yellow River, the Pearl River, and the Huai River.


## 5.2. Solve the Problem

According to Section5.1, four evaluation objects are put forward, and denote as X=x1,x2,x3,x4, where x1,x2,x3,x4 are the Yangtze River, the Yellow River, the Pearl River, and the Huai River, respectively. At the same time, we consider three attributes C=c1,c2,c3, where c1 is monitoring status assessment of the river, c2 is the evaluation of sustainable use of water resources, and c3 is the impact assessment of the hydraulic engineering. In order to evaluate the river health comprehensively, the OLTS and the ILTS are given as follows:(21)OS=s0=very bad,s1=bad,s2=medium,s3=good,s4=very good,IS=n0=river hydrological condition,n1=water environment condition,n2=hydrophytic habitat,n3=water resources development and utilization.Experts express their evaluation information with NPNLTSs. Specifically, they evaluate objects firstly with respect to the OLTS and assess each outer linguistic term in terms of the ILTS. Therefore, a nested probabilistic-numerical linguistic decision matrix is constructed with NPNLTSs, shown as Table1.Table 1
The nested probabilistic-numerical linguistic decision matrix.
c1c2c3x1s00.1n00.2,n10.4,n20.4,s10.1n10.6,n20.4,s20.2n20.2,n30.3,s30.1n30.8s00.2n20.4,s10.5n10.2,n20.2,n20.1,s20.1n20.4,n30.6,s30.2n10.3,n20.2,n30.5s00.2n00.2,n10.4,n20.2,n20.2,s10.2n20.6,n30.4,s20.2n00.4,n10.3,n20.3,s30.2n20.2,n30.8x2s00.5n20.4,s10.2n10.2,n20.3,s20.2n10.2,n20.2,n30.1,s30.1n20.2,n30.8s00.2n00.4,n10.4,s10.3n00.2,n10.2,n20.4,s20.3n10.3,n20.2,n30.3,s30.2n00.1,n10.3,n20.3,n30.3s00.3n10.6,n20.4,s10.2n10.2,n20.3,s20.3n20.1,n30.4,s30.2n20.2,n30.8x3s00.1n00.3,n10.5,s10.2n00.2,n10.4,n20.4,s20.3n20.4,n30.6,s30.4n20.5,n30.5s00.4n00.1,n10.2,n20.2,n30.5,s10.4n10.3,n20.2,s20.1n20.2,n30.8,s30.1n20.4,n30.6s00.2n00.1,n10.3,n20.6,s10.1n10.3,n20.4,n30.3,s20.1n20.2,n30.3,s30.1n20.7,n30.3x4s00.1n20.4,s10.1n10.5,n20.5,s20.1n20.2,n30.3,s30.2n20.4,n30.6s00.1n00.1,n10.4,n20.5,s10.2n10.5,n20.5,s20.1n10.2,n20.2,n30.1,s30.1n10.3,n20.7s00.2n00.1,n10.2,n20.2,s10.2n10.5,n20.5,s20.3n20.2,n30.3,s30.3n10.2,n20.4,n30.4After normalization of attributes, the normalized nested probabilistic-numerical linguistic decision matrix can be shown in Table2.Due to the complexity and uncertainty of the assessment environment, experts give the incompletely weight information about the attributes as follows:(22)0.2≤ω1≤0.4,0.3≤ω2≤0.5,0.2≤ω3≤0.4,0.1ω2≤ω3,ωj≥0,j=1,2,3,∑j=13ωj=1.According to Model I, the optimization model is established to calculate the weight vector of attributes, which is shown as follows:(23)maxDω=∑i=14Diω=∑j=13∑i=14∑k=14ωj4×5∑k=14∑l=15OLijk−OLkjk/4+1+pijk−pkjk+ILijl−ILkjl/5+1+vijl−vkjl4λ1/λs.t.0.2≤ω1≤0.4,0.3≤ω2≤0.5,0.2≤ω3≤0.4,0.1ω2≤ω3,∑j=13ωj=1,ωj≥0,j=1,2,3.Suppose that the parameterλ=1, and the model is simplified as(24)maxDω=0.45ω1+0.5563ω2+0.4525ω3s.t.0.2≤ω1≤0.4,0.3≤ω2≤0.5,0.2≤ω3≤0.4,0.1ω2≤ω3,∑j=13ωj=1,ωj≥0,j=1,2,3.The weight vector can be obtained by Lingo 11, and the result isω=0.2,0.5,0.3T.Next, the proposed extended TOPSIS method is used to evaluate the river health, and the following steps are as follows:Table 2
The normalized nested probabilistic-numerical linguistic decision matrix.
c1c2c3x1s00.2n00.2,n10.4,n20.4,s10.2n10.6,n20.4,s20.4n20.4,n30.6,s30.2n31s00.2n21,s10.5n10.4,n20.4,n30.2,s20.1n20.4,n30.6,s30.2n10.3,n20.2,n30.5s00.25n00.2,n10.4,n20.2,n30.2,s10.25n20.6,n30.4,s20.25n00.4,n10.3,n20.3,s30.25n20.2,n30.8x2s00.5n21,s10.2n10.4,n20.6,s20.2n10.4,n20.4,n30.2,s30.1n20.2,n30.8s00.2n00.5,n10.5,s10.3n00.25,n10.25,n20.5,s20.3n10.375,n20.25,n30.375,s30.2n00.1,n10.3,n20.3,n30.3s00.3n10.6,n20.4,s10.2n10.4,n20.6,s20.3n20.2,n30.8,s30.2n20.2,n30.8x3s00.1n00.375,n10.625,s10.2n00.2,n10.4,n20.4,s20.3n20.4,n30.6,s30.4n20.5,n30.5s00.4n00.1,n10.2,n20.2,n30.5,s10.4n10.6,n20.4,s20.1n20.2,n30.8,s30.1n20.4,n30.6s00.4n00.1,n10.3,n20.6,s10.2n10.3,n20.4,n30.3,s20.2n20.4,n30.6,s30.2n20.7,n30.3x4s00.2n21,s10.2n10.5,n20.5,s20.2n20.4,n30.6,s30.4n20.4,n30.6s00.2n00.1,n10.4,n20.5,s10.4n10.5,n20.5,s20.2n10.4,n20.4,n30.2,s30.2n10.3,n20.7s00.2n00.2,n10.4,n20.4,s10.2n10.5,n20.5,s20.3n20.4,n30.6,s30.3n10.2,n20.4,n30.4Table 3
The positive ideal solutionNPN+ and the negative ideal solution NPN−.
c1c2c3NPN+s00.1n00.375,n10.625,s10.2n00.2,n10.4,n20.4,s20.3n20.4,n30.6,s30.4n20.5,n30.5s00.2n00.5,n10.5,s10.3n00.25,n10.25,n20.5,s20.3n10.375,n20.25,n30.375,s30.2n00.1,n10.3,n20.3,n30.3s00.2n00.2,n10.4,n20.4,s10.2n10.5,n20.5,s20.3n20.4,n30.6,s30.3n10.2,n20.4,n30.4NPN−s00.5n21,s10.2n10.4,n20.6,s20.2n10.4,n20.4,n30.2,s30.1n20.2,n30.8s00.4n00.1,n10.2,n20.2,n30.5,s10.4n10.6,n20.4,s20.1n20.2,n30.8,s30.1n20.4,n30.6s00.4n00.1,n10.3,n20.6,s10.2n10.3,n20.4,n30.3,s20.2n20.4,n30.6,s30.2n20.7,n30.3Step 1. Considering that attributes are all benefit-type, the PISNPN+ and the NIS NPN− of the objects are listed in Table 3 according to Table 2.Step 2. According to equations (15) and (16), the deviation dxi,NPN+ and dxi,NPN− can be calculated as follows:(25)dxi,NPN+=0.0785,0.0452,0.0800,0.0516,dxi,NPN−=0.1224,0.0257,0.0538,0.0727.Step 3. Getdminxi,NPN+=0.0452 and dmaxxi,NPN−=0.1224.Step 4. According to equation (19), the inner weight εixi of the object can be calculated as follows:(26)εixi=0.3430,0.1556,0.2833,0.2182T.Step 5. According to equation (20), the inner-layer score IZi of the object can be calculated as follows:(27)IZ1=0.12530.25050.34940.0567000.26800.36940.144400.11330.08500.28440.2991000.04670.08780.64740,IZ2=0.07780.24770.4563000.03890.28940.45350000.19550.23270.353600.01560.04670.17190.54770,IZ3=0.17250.33050.20110.077800.06860.31550.31270.08500000.28160.50020000.43200.34980,IZ4=0.07220.17550.53410000.39090.39090000.06220.31270.4069000.10330.35940.31910.Step 6. According to Definition2, the whole score FNPNi of the object are(28)FNPNi=0.8071,0.7602,0.7227,0.7653.Step 7. Rank the objects by comparison rule, that is,x1>x4>x2>x3, and the best object is the Yangtze River.


## 5.3. Comparative Analysis and Discussion

In order to verify the superiority of the proposed method, we make some comparative analysis based on simulation experiments by MATLAB from three aspects: (1) the impact for the results without weight optimization; (2) the impact for the results under other uncertain environments; (3) the impact for the results using other decision-making methods.


### 5.3.1. The Impact without Weight Optimization

Suppose that the attribute weights satisfy several cases, such as uniform distribution, increase progressively, decrease progressively, and irregular situation. With different attribute weights, the whole scores of the objects and the rankings are shown in Table4.Table 4
The whole scores of the objects and the rankings with different attribute weights.
ωFx1Fx2Fx3Fx4Rankingsω=1/3,1/3,1/3T0.74440.70080.63880.6921x1>x2>x4>x3ω=0.1,0.3,0.6T0.78330.73090.68190.7318x1>x4>x2>x3ω=0.6,0.3,0.1T0.69630.66020.58030.6396x1>x4>x2>x3ω=0.4,0.5,0.1T0.76560.72480.67380.7207x1>x2>x4>x3ω=0.2,0.5,0.3T0.80710.76020.72270.7653x1>x4>x2>x3As we can see, different attribute weights have effect on the whole scores of the objects, and the rankings are also a little different. Although the best object and the worst object arex1 and x3, respectively, the rank of x2 and x4 are different because of the various attribute weights. Therefore, it can be concluded that when the numbers of objects and attributes are large, it is necessary to make full use of information to optimize the weight. Because various attribute weights may lead to different ranking results, it is also one of the most important factors to obtain a rational result.In the following, we further analyze the relationship between the attribute weights and the whole scores of objects. We generate 1000 sets of attribute weights randomly based on MATLAB and apply to the same case, and the whole scores of each object can be shown in Figure2.Figure 2
The whole scores of each object with 1000 sets of attribute weights: (a) two-dimensional diagram; (b) three-dimensional diagram.
(a)(b)In general, the range of the whole scores is from 0.5 to 0.95. In Figure2(a), the whole scores of x1 are almost the highest, while the whole scores of x3 are almost the lowest. And the whole scores of x2 and x4 are larger and smaller alternatively. In Figure 2(b), we can see the trend clearly with various attribute weights. Specifically, the maximum values of the whole scores of four objectives are 0.9384, 0.8950, 0.8942, and 0.9167, respectively. The minimum values of the whole scores of four objectives are 0.6623, 0.6234, 0.5064, and 0.5857, respectively. Therefore, the attribute weights can affect the whole scores about 20%, and it is meaningful to determine the weights rationally and scientifically.


### 5.3.2. The Impact under Other Uncertain Environments

Next, we study the case using different linguistic evaluation methods. In Section5.2, experts use the NPNLTSs to express the evaluation information considering outer and inner linguistic information. Then, hesitant fuzzy linguistic term sets (HFLTSs) [8] and probabilistic linguistic term sets (PLTSs) [9] are used to assess evaluation information of the objects with respect to the attributes, respectively. Table 5 and 6 list the evaluation information with HFLTSs and PLTSs, respectively.Table 5
The evaluation information with HFLTSs.
c1c2c3x1s2,s3s1,s2,s3s0,s1,s2,s3x2s2s1,s2,s3s2,s3x3s2,s3s0,s1,s2,s3s0,s1,s2x4s2,s3s1,s2s1,s2,s3Table 6
The evaluation information with PLTSs.
c1c2c3x1s00.2,s10.2,s20.4,s30.2s00.2,s10.5,s20.1,s30.2s00.25,s10.25,s20.25,s30.25x2s00.5,s10.2,s20.2,s30.1s00.2,s10.3,s20.3,s30.2s00.3,s10.2,s20.3,s30.2x3s00.1,s10.2,s20.3,s30.4s00.4,s10.4,s20.1,s30.1s00.4,s10.2,s20.2,s30.2x4s00.2,s10.2,s20.2,s30.4s00.2,s10.4,s20.2,s30.2s00.2,s10.2,s20.3,s30.3Similarly, we use TOPSIS methods with HFLTSs and PLTSs to deal with the case, and the rankings are listed in Table7.Table 7
Rankings with different linguistic models.
MethodsRankingsThe TOPSIS method with HFLTSs [40]x4>x1>x3>x2The TOPSIS method with PLTSs [9]x1>x2>x4>x3The proposed methodx1>x4>x2>x3The ranking results show thatx1 is the best object by TOPSIS methods with PLTSs and NPNLTSs, while x4 is the best object by TOPSIS methods with HFLTSs, and the rankings are different by TOPSIS methods with three various linguistic models. The reason may be that HFLTSs do not consider the weight of linguistic terms, and PLTSs cannot better reflect information comprehensively according to Tables 5 and 6. Therefore, the results by the decision method with NPNLTSs are more reasonable and reliable.


### 5.3.3. The Impact Using Other Decision-Making Methods

Finally, we make analysis to compare the proposed method with other decision-making methods with NPNLTSs, which are aggregation method [18], VIKOR method [41], and ELECTRE method [42]. Suppose that the attribute weights are ω=0.2,0.5,0.3T, four methods above are used to deal with the same case, and the ranking results are listed in Table 8.Table 8
Rankings by four decision-making methods.
MethodsRankingsThe aggregation method [18]x1>x4>x3>x2The VIKOR method [41]x1>x4>x2>x3The ELECTRE method [42]x1>x2>x4>x3The proposed methodx1>x4>x2>x3As we can see from Table8, ranking results by four decision-making methods are a little different. Specifically, all the best objects are x1, and the rankings are the same by the proposed method and the VIKOR method. To some extent, the proposed TOPSIS method is reliable.In order to further study the effectiveness of the proposed method, we make simulation experiments to analyze the average operation time (AOT) by four decision-making methods. Letm and k be the number of objects and attributes, respectively. When m and k are from 3 to 12 after 1000 simulation times, the AOTs by four decision-making methods are shown in Figure 3.Figure 3
The average operation time by four decision-making methods: (a)m is from 3 to 12 with k=3; (b) k is from 3 to 12 with m=3.
(a)(b)Whenm and k are from 3 to 12, AOTs with four decision-making methods are all gradually increasing, and there are apparent differences by various methods. In Figure 3(a), the aggregation method takes the minimum AOT, and the ELECTRE method takes the maximum AOT. The AOT of the proposed method is little more than the aggregation method. In Figure 3(b), the minimum AOT also belongs to the aggregation method, while the maximum AOT belongs to the ELECTRE method. The proposed method also takes more time than the aggregation method. In addition, increasing the number of objects would take more time than increasing the number of attributes. Therefore, the proposed method is effective without taking too much time, and the results by the proposed method are also reliable.


## 5.3.1. The Impact without Weight Optimization

Suppose that the attribute weights satisfy several cases, such as uniform distribution, increase progressively, decrease progressively, and irregular situation. With different attribute weights, the whole scores of the objects and the rankings are shown in Table4.Table 4
The whole scores of the objects and the rankings with different attribute weights.
ωFx1Fx2Fx3Fx4Rankingsω=1/3,1/3,1/3T0.74440.70080.63880.6921x1>x2>x4>x3ω=0.1,0.3,0.6T0.78330.73090.68190.7318x1>x4>x2>x3ω=0.6,0.3,0.1T0.69630.66020.58030.6396x1>x4>x2>x3ω=0.4,0.5,0.1T0.76560.72480.67380.7207x1>x2>x4>x3ω=0.2,0.5,0.3T0.80710.76020.72270.7653x1>x4>x2>x3As we can see, different attribute weights have effect on the whole scores of the objects, and the rankings are also a little different. Although the best object and the worst object arex1 and x3, respectively, the rank of x2 and x4 are different because of the various attribute weights. Therefore, it can be concluded that when the numbers of objects and attributes are large, it is necessary to make full use of information to optimize the weight. Because various attribute weights may lead to different ranking results, it is also one of the most important factors to obtain a rational result.In the following, we further analyze the relationship between the attribute weights and the whole scores of objects. We generate 1000 sets of attribute weights randomly based on MATLAB and apply to the same case, and the whole scores of each object can be shown in Figure2.Figure 2
The whole scores of each object with 1000 sets of attribute weights: (a) two-dimensional diagram; (b) three-dimensional diagram.
(a)(b)In general, the range of the whole scores is from 0.5 to 0.95. In Figure2(a), the whole scores of x1 are almost the highest, while the whole scores of x3 are almost the lowest. And the whole scores of x2 and x4 are larger and smaller alternatively. In Figure 2(b), we can see the trend clearly with various attribute weights. Specifically, the maximum values of the whole scores of four objectives are 0.9384, 0.8950, 0.8942, and 0.9167, respectively. The minimum values of the whole scores of four objectives are 0.6623, 0.6234, 0.5064, and 0.5857, respectively. Therefore, the attribute weights can affect the whole scores about 20%, and it is meaningful to determine the weights rationally and scientifically.


## 5.3.2. The Impact under Other Uncertain Environments

Next, we study the case using different linguistic evaluation methods. In Section5.2, experts use the NPNLTSs to express the evaluation information considering outer and inner linguistic information. Then, hesitant fuzzy linguistic term sets (HFLTSs) [8] and probabilistic linguistic term sets (PLTSs) [9] are used to assess evaluation information of the objects with respect to the attributes, respectively. Table 5 and 6 list the evaluation information with HFLTSs and PLTSs, respectively.Table 5
The evaluation information with HFLTSs.
c1c2c3x1s2,s3s1,s2,s3s0,s1,s2,s3x2s2s1,s2,s3s2,s3x3s2,s3s0,s1,s2,s3s0,s1,s2x4s2,s3s1,s2s1,s2,s3Table 6
The evaluation information with PLTSs.
c1c2c3x1s00.2,s10.2,s20.4,s30.2s00.2,s10.5,s20.1,s30.2s00.25,s10.25,s20.25,s30.25x2s00.5,s10.2,s20.2,s30.1s00.2,s10.3,s20.3,s30.2s00.3,s10.2,s20.3,s30.2x3s00.1,s10.2,s20.3,s30.4s00.4,s10.4,s20.1,s30.1s00.4,s10.2,s20.2,s30.2x4s00.2,s10.2,s20.2,s30.4s00.2,s10.4,s20.2,s30.2s00.2,s10.2,s20.3,s30.3Similarly, we use TOPSIS methods with HFLTSs and PLTSs to deal with the case, and the rankings are listed in Table7.Table 7
Rankings with different linguistic models.
MethodsRankingsThe TOPSIS method with HFLTSs [40]x4>x1>x3>x2The TOPSIS method with PLTSs [9]x1>x2>x4>x3The proposed methodx1>x4>x2>x3The ranking results show thatx1 is the best object by TOPSIS methods with PLTSs and NPNLTSs, while x4 is the best object by TOPSIS methods with HFLTSs, and the rankings are different by TOPSIS methods with three various linguistic models. The reason may be that HFLTSs do not consider the weight of linguistic terms, and PLTSs cannot better reflect information comprehensively according to Tables 5 and 6. Therefore, the results by the decision method with NPNLTSs are more reasonable and reliable.


## 5.3.3. The Impact Using Other Decision-Making Methods

Finally, we make analysis to compare the proposed method with other decision-making methods with NPNLTSs, which are aggregation method [18], VIKOR method [41], and ELECTRE method [42]. Suppose that the attribute weights are ω=0.2,0.5,0.3T, four methods above are used to deal with the same case, and the ranking results are listed in Table 8.Table 8
Rankings by four decision-making methods.
MethodsRankingsThe aggregation method [18]x1>x4>x3>x2The VIKOR method [41]x1>x4>x2>x3The ELECTRE method [42]x1>x2>x4>x3The proposed methodx1>x4>x2>x3As we can see from Table8, ranking results by four decision-making methods are a little different. Specifically, all the best objects are x1, and the rankings are the same by the proposed method and the VIKOR method. To some extent, the proposed TOPSIS method is reliable.In order to further study the effectiveness of the proposed method, we make simulation experiments to analyze the average operation time (AOT) by four decision-making methods. Letm and k be the number of objects and attributes, respectively. When m and k are from 3 to 12 after 1000 simulation times, the AOTs by four decision-making methods are shown in Figure 3.Figure 3
The average operation time by four decision-making methods: (a)m is from 3 to 12 with k=3; (b) k is from 3 to 12 with m=3.
(a)(b)Whenm and k are from 3 to 12, AOTs with four decision-making methods are all gradually increasing, and there are apparent differences by various methods. In Figure 3(a), the aggregation method takes the minimum AOT, and the ELECTRE method takes the maximum AOT. The AOT of the proposed method is little more than the aggregation method. In Figure 3(b), the minimum AOT also belongs to the aggregation method, while the maximum AOT belongs to the ELECTRE method. The proposed method also takes more time than the aggregation method. In addition, increasing the number of objects would take more time than increasing the number of attributes. Therefore, the proposed method is effective without taking too much time, and the results by the proposed method are also reliable.


## 6. Conclusions

In the complex environment, nested probabilistic-numerical linguistic term sets (NPNLTSs) are useful and flexible tools to express qualitative and quantitative information. Considering that attribute weight information is often incomplete and uncertain in decision-making problems. In this paper, we have established an optimization model with NPNLTSs based on distance measures to obtain the attribute weight. Combined with the TOPSIS method and NPNLTSs, we have proposed an optimization-based TOPSIS method to deal with complex and uncertain decision-making problems, and have developed the corresponding algorithm. A case study about the river health assessment has been presented to show the effectiveness and practicability of the proposed method. In addition, we have made comparative analysis from three aspects including the impact for the results without weight optimization, the impact for the results under other uncertain environments, and the impact for the results using other decision-making methods. After discussions, the optimization-based TOPSIS method proposed in this paper shows effective and reliable, and the results make a technically sound contribution to the decision-making field.In the future, some interesting topics about decision-making methods with NPNLTSs will be further studied. For example, the decision-making methods based on preference relation can be investigated with incomplete information. Moreover, we will study the situation where attribute weight and evaluation information are both incomplete and establish a rational decision-making method.



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*Source: 5092531-2020-07-31.xml*