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Volume 12, Issue 1, November 2018, Pages 410 - 425

A Pythagorean Fuzzy TOPSIS Method Based on Novel Correlation Measures and Its Application to Multiple Criteria Decision Analysis of Inpatient Stroke Rehabilitation

Authors
Yu-Li Lin1, Lun-Hui Ho2, Shu-Ling Yeh3, Ting-Yu Chen4, *
1Department of Nursing, Linkou Chang Gung Memorial Hospital, Department of Nursing, Chang Gung University of Science and Technology, No.5, Fuxing St., Guishan District, Taoyuan City 333, Taiwan
2Department of Nursing, Linkou Chang Gung Memorial Hospital, Department of Nursing, Chang Gung University of Science and Technology, No.5, Fuxing St., Guishan District, Taoyuan City 333, Taiwan
3Department of Nursing, Linkou Chang Gung Memorial Hospital, Department of Nursing, Chang Gung University of Science and Technology, No.5, Fuxing St., Guishan District, Taoyuan City 333, Taiwan
4Graduate Institute of Business and Management, Chang Gung University, Department of Industrial and Business Management, Chang Gung University, Department of Nursing, Linkou Chang Gung Memorial Hospital, No. 259, Wenhua 1st Rd., Guishan District, Taoyuan City 33302, Taiwan
*

Corresponding author. Email: tychen@mail.cgu.edu.tw

Received 29 May 2018, Revised 15 October 2018, Accepted 24 October 2018, Available Online 1 November 2018.
DOI
10.2991/ijcis.2018.125905657How to use a DOI?
Keywords
Pythagorean fuzzy set; Multiple criteria decision analysis; TOPSIS; Correlation measure; Inpatient stroke rehabilitation
Abstract

The complex nature of the realistic decision-making process requires the use of Pythagorean fuzzy (PF) sets which have been shown to be a highly promising tool capable of solving highly vague and imprecise problems. Multiple criteria decision analysis (MCDA) methods within the PF environment are very attractive approaches for today’s intricate decision environments. With this study, an effective compromise model named as the PF technique for order preference by similarity to ideal solutions (TOPSIS) is proposed based on some novel PF correlation-based concepts to overcome the complexities and ambiguities involved in real-life decision situations. In contrast to the existing distance-based definitions, this paper develops new closeness indices based on an extended concept of PF correlations. This paper employs the proposed PF correlation coefficients to construct two types of closeness measures. A comprehensive concept of PF correlation-based closeness indices can then be established to balance the consequences yielded by the two closeness measures. Based on these useful concepts, an effective PF TOPSIS method is proposed to address MCDA problems involving PF information and determine the ultimate priority orders among competing alternatives. Feasibility and practicability of the developed approach are illustrated by a medical decision-making problem of inpatient stroke rehabilitation. Finally, the proposed methodology is compared with other current methods to further explain its effectiveness.

Copyright
© 2019 The Authors. Published by Atlantis Press SARL.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

Multiple criteria decision analysis (MCDA) concerns about evaluating discrete candidate alternatives and selecting the best compromise solution among a finite set of alternatives based on a finite set of criteria. There are numerous MCDA methods proposed by researchers in literature [1, 2]. The technique for order preference by similarity to ideal solutions (TOPSIS), initiated by Hwang and Yoon [3] and later extended by Yoon [4] and Hwang et al. [5], is the most widely used compromise model in the MCDA field. TOPSIS ranks alternatives and determines the compromise solution that is the closest to the ideal solution. More precisely, the rationale of classical TOPSIS methods is that the best compromise alternative should have the shortest distance from the positive-ideal solution and the longest distance from the negative-ideal solution [1, 6, 7]. TOPSIS has hitherto been widely studied by researchers and practitioners and have been successfully applied to several fields of real decision-makings [1, 7–11].

Regarding the uncertainty in real situations, many fuzzy extensions related studies have been explored to enrich the theory of TOPSIS methodology [1, 10, 12]. It is noteworthy that numerous realistic MCDA problems involve risks and uncertainties in nature [2, 13]. In this regard, the fuzzy set theory is very suitable and effective to handle the MCDA problems under vague, uncertain, and incomplete information environment. From this perspective, different versions of TOPSIS based on fuzzy sets have been developed for considering uncertainties and vagueness in MCDA problems, such as the fuzzy TOPSIS [14, 15], the generalized fuzzy TOPSIS [1], the analytic network process weighted fuzzy TOPSIS [16], the intuitionistic fuzzy TOPSIS [7, 9, 17], the interval-valued intuitionistic fuzzy linguistic TOPSIS [18], the interval type-2 fuzzy TOPSIS [10, 12], and so on. Although numerous papers with fuzzy sets have been proposed for developing extensions of TOPSIS and applied to different application areas, relatively little attention has been paid to the extended TOPSIS dealing with MCDA problems under complex uncertainty based on Pythagorean fuzzy (PF) sets.

The concept of PF sets was introduced by Yager [19] and Yager and Abbasov [20]. PF sets are characterized by flexible degrees of membership, non-membership, and indeterminacy, in which the square sum of the degree of membership and the degree of nonmembership is less than or equal to one [19–21]. Since Zhang and Xu [22] proposed the general mathematical forms of PF sets, the PF theory has become increasingly popular in the MCDA field [23–25]. In particular, PF sets relax the constraint conditions and possess a great capability of managing high-order uncertainty in real-world decision situations [21, 23, 24]. Accordingly, many researchers have studied the MCDA methods within the PF decision environment [21, 24–31], and recently their popularity has grown among scholars owing to their high level of effectiveness [30, 31]. Nevertheless, relatively few studies focus on the development of the PF TOPSIS methodology.

Zhang and Xu [22] extended the TOPSIS method to effectively deal with the MCDA problems with PF sets and employed the revised closeness to identify the optimal alternative. Zeng et al. [32] combined the weighted average and ordered weighted averaging operator with distance measures to construct a PF- ordered weighted averaging weighted average distance operator and develop a hybrid TOPSIS method. Under the PF uncertainty, Liang and Xu [33] proposed a new concept of hesitant PF sets and explored their application to MCDA with the aid of the TOPSIS method. In particular, the three papers mentioned here employed the PF distance metrics as the separation measures to determine the degrees of relative closeness (or closeness coefficients) required in their proposed TOPSIS procedures. Based on the vertex method via Euclidean distances, Gul and Ak [34] developed a PF TOPSIS method to assess the hazards with respect to the parameters of likelihood and severity. By using the Hamming distance measure, Liang et al. [35] adopted the TOPSIS technique to estimate the conditional probability and propose a method for three-way decisions using ideal TOPSIS solutions at PF information. Zhou et al. [36] resembled the TOPSIS method, which considers the symmetry of the distances to the positive- and negative-ideal solutions, into their multiple criteria group decision-making method based on the Pythagorean normal cloud.

As is well known, the main approach in the classical TOPSIS procedure is to take the most preferred alternative which has the (weighted) minimum distance to the positive-ideal solution and the (weighted) maximum distance to the negative-ideal solution in a geometrical sense [3, 5]. Central to the TOPSIS procedure is the relative closeness with respect to the ideal solutions. Accordingly, most existing studies on the TOPSIS models and techniques have focused on distance-based separation measures for determining the relative closeness (or closeness coefficients) and then ranking the preference orders among alternatives. Analogously, to solve MCDA problems within the PF environment, Gul and Ak [34], Liang and Xu [33], Liang et al. [35], Zeng et al. [32], Zhang and Xu [22], and Zhou et al. [36] also employed distance measures to determine degrees of relative closeness or revised closeness. Nevertheless, except for distance-based separation measures, the PF TOPSIS and other versions of TOPSIS extensions have not been yet sufficiently investigated for real-world MCDA problems in the PF context, which motivates the research of this paper.

This paper aims to present a useful extension of TOPSIS using novel PF correlation-based closeness indices and develops an effective PF TOPSIS method for managing MCDA problems under complex PF uncertainty. Different from the distance-based separation measures and traditional relative closeness, this paper defines a new closeness index based on the extended concept of PF correlations. The proposed PF correlations can fully reflect the relationship between PF information. By conducting a PF correlation analysis, the interdependency of an alternative with respect to the positive- and negative-ideal PF solutions can be appropriately examined with the aid of new closeness measures and PF correlation-based closeness indices. More specifically, this paper provides new definitions of (weighted) PF correlation coefficients for the purpose of developing certain useful concepts of (weighted) Type I and Type II closeness measures. Next, this paper constructs a comprehensive concept of (weighted) PF correlation-based closeness indices to acquire a balanced consequence between the obtained results via the (weighted) Type I and Type II closeness measures. A simple and effective PF TOPSIS method is then established to address MCDA problems involving PF information and further determine the ultimate priority orders of alternatives.

Finally, a practical medical decision-making problem concerning rehabilitation treatments for hospitalized patients with stroke and cerebrovascular diseases is provided to illustrate the effectiveness and feasibility of the proposed PF TOPSIS methodology. Stroke rehabilitation substantially contributes to the prevention of relapse as well as to patients’ recovery, adaption to disability, and quality of life. This real-world application focuses on the treatment of patients with acute stroke and utilizes the PF TOPSIS method to evaluate the priority of various rehabilitation treatment measures for hospitalized patients. The application results can provide a useful decision-aiding suggestion for medical practitioners.

The remainder of this paper is organized as follows: Section 2 reviews some basic concepts related to PF sets that are used throughout this paper. Section 3 formulates an MCDA problem based on PF sets and presents the concept of the positive- and negative-ideal PF solutions. Section 4 introduces novel correlation measures named as the (weighted) PF correlation coefficients and explores their essential properties. Section 5 presents new PF correlation-based closeness indices and develops an effective PF TOPSIS method for managing MCDA problems within the PF decision environment. Section 6 applies the proposed method and techniques to address a practical medical decision-making problem concerning hospitalization rehabilitation treatments for stroke patients to demonstrate its feasibility and applicability. Finally, Section 7 presents the conclusions.

2. PRELIMINARY DEFINITIONS

This section introduces the basic concepts of PF sets and presents some arithmetic operations related to PF information.

Definition 1.

[19, 20, 22, 24, 28] A PF set in a finite universe of discourse X is an object having the following form:

P=〈x,μP(x),νP(x)〉|x∈X,
where μP (x): X → [0, 1] and νP (x): X → [0, 1] denote, respectively, the degree of membership and the degree of nonmembership with the condition:
μPx2+νPx2≤1
for each element x ∈ X. Let p = (μP(x), νP(x)) denote a PF value. The degree of indeterminacy relative to P for each x ∈ X is defined as follows:
πPx=1−μPx2−νPx2.
The degree πP(x) expresses a lack of knowledge of whether the element x belongs to P or not.

Definition 2.

[24, 26–28] Let p1, p2, and p be three PF values in X and α ≥ 0. Some basic arithmetic operations are defined as follows:

p1∨p2=maxμP1x,μP2x,minνP1x,νP2x,
p1∧p2=minμP1x,μP2x,maxνP1x,νP2x
p1⊕p2=μP1x2+μP2x2−μP1x2⋅μP2x20.5,νP1x⋅νP2x
p1⊗p2=μP1x⋅μP2x,νP1x2+νP2x2−νP1(x)2⋅νP2(x)20.5,
α⋅p=1−1−μPx2α,νPxα,
pα=μPxα,1−1−νPx2α.

3. DESCRIPTION OF PF MCDA PROBLEMS

This section first describes an MCDA problem under complex uncertainty based on PF sets. Next, this section identifies the positive- and negative-ideal PF solutions as points of reference within the PF environment.

Consider an MCDA problem that contains a discrete set of m (m ≥ 2) candidate alternatives, expressed as Z = {z1, z2, ⋯, zm}. Let C = {c1, c2, ⋯, cn} be a finite set of n (n ≥ 2) evaluative criteria that have the weight vector w = (w1, w2, ⋯, wn), where wj ∈ [0, 1] for all j ∈ {1, 2, ⋯, n} and ∑j=1nwj=1. Set C can be generally divided into two sets, CB and CC, where CB denotes a collection of benefit criteria (i.e., larger values of cj indicate a greater preference), and CC denotes a collection of cost criteria (i.e., smaller values of cj indicate a greater preference). Moreover, CB ∩ CC = ∅ and CB ∪ CC = C.

In the PF context, the MCDA problem consisting of PF values can be concisely represented in the following matrix form:

p=pijm×n=(μ11,ν11)(μ12,ν12)⋯(μ1n,ν1n)(μ21,ν21)(μ22,ν22)⋯(μ2n,ν2n)⋮⋮⋱⋮(μm1,νm1)(μm2,νm2)⋯(μmn,νmn).
The element pij = (μij, νij) in the PF decision matrix p indicates the evaluative rating of an alternative zi ∈ Z with respect to a criterion cj ∈ C, where the degree of membership μij and the degree of nonmembership νij fulfill μij ∈ [0, 1], νij ∈ [0, 1], and (μij)2 + (νij)2 ≤ 1. The degree of indeterminacy that corresponds to each pij is given by πij=1−μij2−νij2. Furthermore, the PF characteristic Pi of an alternative zi can be represented by all of the corresponding PF values as follows:
Pi=〈c1,pi1〉,〈c2,pi2〉,⋯,〈cn,pin〉=〈c1,(μi1,νi1)〉,〈c2,(μi2,νi2)〉,⋯,〈cn,(μin,νin)〉.

Definition 3

Let z+ and z− denote the positive- and negative-ideal PF solutions, respectively, with respect to a PF decision matrix p = [(μij, νij)] m×n. The PF characteristics of z+ and z− are represented as follows:

P+=〈c1,p+1〉,〈c2,p+2〉,⋯,〈cn,p+n〉,
P−=〈c1,p−1〉,〈c2,p−2〉,⋯,〈cn,p−n〉,
where the positive- and negative-ideal PF values within P+ and P− are defined as follows:
p+j=μ+j,ν+j=∨i=1mpij=maxi=1mμij,mini=1mνijif  cj∈CB,∧i=1mpij=mini=1mμij,maxi=1mνijif  cj∈CC,
p−j=μ−j,ν−j=∧i=1mpij=mini=1mμij,maxi=1mνijif  cj∈CB,∨i=1mpij=maxi=1mμij,mini=1mνijif  cj∈CC.
The respective degrees of indeterminacy corresponding to p+j and p−j are given by π+j=1−μ+j2−ν+j2 and π−j=1−μ−j2−ν−j2.

4. NOVEL PF CORRELATION COEFFICIENTS

In the PF decision environment, this section attempts to develop new correlation measures named as the PF correlation coefficient and the weighted PF correlation coefficient. Some desirable and useful properties of these new measures are also investigated in this section. The (weighted) PF correlation coefficients can facilitate expressing not only a relative strength but also a positive or negative relationship between the two PF characteristics.

Definition 4

Let Pi1 and Pi2 be two PF characteristics in the PF decision matrix p, where Pi1=〈c1,(μi11,νi11)〉,〈c2,(μi12,νi12)〉,⋯,〈cn,(μi1n,νi1n)〉 and Pi2=〈c1,(μi21,νi21)〉,〈c2,(μi22,νi22)〉,⋯,〈cn,(μi2n,νi2n)〉. Let μ¯j=∑i=1mμij/m, ν¯j=∑i=1mνij/m, and π¯j=∑i=1mπij/m. The PF correlation coefficient γ(Pi1,Pi2) between Pi1 and Pi2 is defined as follows:

γPi1,Pi2=13rμPi1,Pi2+rνPi1,Pi2+rπPi1,Pi2,
where
rμPi1,Pi2=∑j=1nμi1j2−μ¯j2⋅μi2j2−μ¯j2∑j=1nμi1j2−μ¯j22⋅∑j=1nμi2j2−μ¯j22,
rνPi1,Pi2=∑j=1nνi1j2−ν¯j2⋅νi2j2−ν¯j2∑j=1nνi1j2−ν¯j22⋅∑j=1nνi2j2−ν¯j22,
rπPi1,Pi2=∑j=1nπi1j2−π¯j2⋅πi2j2−π¯j2∑j=1nπi1j2−π¯j22⋅∑j=1nπi2j2−π¯j22.

It is worth noting that this paper avoids zero in the denominators of the membership component rμPi1,Pi2, the nonmembership component rνPi1,Pi2, and the indeterminacy component rπPi1,Pi2 with respect to each γPi1,Pi2 without loss of generality.

Theorem 1.

The membership component rμPi1,Pi2 in the PF correlation coefficient γPi1,Pi2 for two PF characteristics Pi1 and Pi2 satisfies the following properties:

  • (T1.1) rμPi1,Pi2=rμPi2,Pi1;

  • (T1.2) rμPi1,Pi2=1 if μi1j=μi2j for all cj ∈ C;

  • (T1.3) |rμPi1,Pi2|≤1.

Proof.

(T1.1) is trivial. (T1.2) can be easily checked because:

rμPi1,Pi2=∑j=1nμi2j2−μ¯j22∑j=1nμi2j2−μ¯j222=1.

For (T1.3), it is clearly known that 0 ≤ (μij)2 ≤ 1 and 0≤μ¯j2≤1 because μij ∈ [0, 1] and μ¯j=∑i=1mμij/m. Hence, it follows that −1≤μi1j2−μ¯j2≤1 and −1≤μi2j2−μ¯j2≤1. One can easily infer that 0≤μi1j2−μ¯j22≤1 and 0≤μi2j2−μ¯j22≤1, which lead to −1≤μi1j2−μ¯j2⋅μi2j2−μ¯j2≤1. Thus, the range of the numerator in Equation (17) will be −n≤∑j=1nμi1j2−μ¯j2⋅μi2j2−μ¯j2≤n. Next, consider the denominator in Equation (17). By means of 0≤μi1j2−μ¯j22≤1 and 0≤μi2j2−μ¯j22≤1, the following results are correct: 0≤∑j=1nμi1j2−μ¯j22≤n and 0≤∑j=1nμi2j2−μ¯j22≤n. Thus, ∑j=1nμi1j2−μ¯j22⋅∑j=1nμi2j2−μ¯j22≤n⋅ n=n. Consequently, it can be concluded that −1≤rμPi1,Pi2≤1, that is, |rμPi1,Pi2|≤1. This establishes the theorem.

Theorem 2.

The nonmembership component rνPi1,Pi2 in the PF correlation coefficient γPi1,Pi2 for two PF characteristics Pi1 and Pi2 satisfies the following properties:

  • (T2.1) rνPi1,Pi2=rνPi2,Pi1;

  • (T2.2) rνPi1,Pi2=1 if νi1j=νi2j for all cj ∈ C;

  • (T2.3) |rνPi1,Pi2|≤1.

Proof.

The proofs of this theorem are analogous to those of Theorem 1.

Theorem 3.

The indeterminacy component rπPi1,Pi2 in the PF correlation coefficient γPi1,Pi2 for two PF characteristics Pi1 and Pi2 satisfies the following properties:

  • (T3.1) rπPi1,Pi2=rπPi2,Pi1;

  • (T3.2) rπPi1,Pi2=1 if πi1j=πi2j for all cj ∈ C;

  • (T3.3) |rπPi1,Pi2|≤1.

Proof.

The proofs of this theorem are analogous to those of Theorem 1.

Theorem 4.

The PF correlation coefficient γPi1,Pi2 between two PF characteristics Pi1 and Pi2 satisfies the following properties:

  • (T4.1) γPi1,Pi2=γPi2,Pi1;

  • (T4.2) γPi1,Pi2=1 if Pi1=Pi2;

  • (T4.3) |γPi1,Pi2|≤1.

Proof.

(T4.1) is straightforward from (T1.1), (T2.1), and (T3.1). For (T4.2), the assumption Pi1=Pi2 means that μi1j=μi2j, νi1j=νi2j, and πi1j=πi2j hold for all cj ∈ C. According to (T1.2), (T2.2), and (T3.2), one obtains rμPi1,Pi2=rνPi1,Pi2=rπPi1,Pi2=1, which brings about γPi1,Pi2=1. Thus, (T4.2) is valid. For (T4.3), based on the results of (T1.3), (T2.3), and (T3.3), one can infer that |γPi1,Pi2|≤1, i.e., (T4.3) is valid. This completes the proof.

Furthermore, this paper incorporates the weight vector w = (w1, w2, ⋯, wn) into the correlation measure to propose the weighted PF correlation coefficient between two PF characteristics.

Definition 5.

Let Pi1 and Pi2 be two PF characteristics in the PF decision matrix p, and let wj be the importance weight of criterion cj ∈ C with wj ∈ [0, 1] and ∑j=1nwj=1. The weighted PF correlation coefficient γwPi1,Pi2 between Pi1 and Pi2 is defined as follows:

γwPi1,Pi2=13rμwPi1,Pi2+rνwPi1,Pi2+rπwPi1,Pi2,
where
rμwPi1,Pi2=∑j=1nwjμi1j2−μ¯j2⋅μi2j2−μ¯j2/∑j=1nwjμi1j2−μ¯j22⋅∑j=1nwjμi2j2−μ¯j22,
rνwPi1,Pi2=∑j=1nwjνi1j2−ν¯j2⋅νi2j2−ν¯j2/∑j=1nwjνi1j2−ν¯j22⋅∑j=1nwjνi2j2−ν¯j22,
rπwPi1,Pi2=∑j=1nwjπi1j2−π¯j2⋅πi2j2−π¯j2/∑j=1nwjπi1j2−π¯j22⋅∑j=1nwjπi2j2−π¯j22.

Analogous to the unweighted situation, this paper avoids zero in the denominators of the membership component rμwPi1,Pi2, the nonmembership component rνwPi1,Pi2, and the indeterminacy component rπwPi1,Pi2 with respect to each γwPi1,Pi2 without loss of generality.

Theorem 5.

The membership component rμwPi1,Pi2 in the weighted PF correlation coefficient γwPi1,Pi2 for two PF characteristics Pi1 and Pi2 satisfies the following properties:

  • (T5.1) rμwPi1,Pi2=rμwPi2,Pi1;

  • (T5.2) rμwPi1,Pi2=1 if μi1j=μi2j for all cj ∈ C;

  • (T5.3) |rμwPi1,Pi2|≤1;

  • (T5.4) rμwPi1,Pi2=rμPi1,Pi2 if w = (1/n, 1/n, ⋯, 1/n).

Proof.

(T5.1) is trivial. (T5.2) is obvious because:

rμwPi1,Pi2=∑j=1nwjμi2j2−μ¯j22/∑j=1nwjμi2j2−μ¯j222=1.
For (T5.3), based on the previous discussion in the proving process of (T1.3), one obtains −1≤μi1j2−μ¯j2⋅μi2j2−μ¯j2≤1. Because ∑j=1nwj=1, the range of the numerator in Equation (21) will be −1≤∑j=1nwjμi1j2−μ¯j2⋅μi2j2−μ¯j2≤1. Consider the component in the denominator of Equation (21). Because 0≤∑j=1nμi1j2−μ¯j22≤n and 0≤∑j=1nμi2j2−μ¯j22≤n based on the obtained results in the proving process of (T1.3), one has 0≤∑j=1nwjμi1j2−μ¯j22≤1 and 0≤∑j=1nwjμi2j2−μ¯j22≤1. If follows that 0≤∑j=1nwjμi1j2−μ¯j221/2⋅∑j=1nwjμi2j2−μ¯j221/2≤1. Recall that this paper avoids zero in the denominator of rμwPi1,Pi2. Hence, (T5.3) is valid because −1≤rμwPi1,Pi2≤1i.e. |rμwPi1,Pi2|≤1. For (T5.4), when the weight vector w = (1/n, 1/n, ⋯, 1/n), the membership component rμwPi1,Pi2 becomes:
rμwPi1,Pi2=∑j=1n1nμi1j2−μ¯j2⋅μi2j2−μ¯j2/∑j=1n1nμi1j2−μ¯j22⋅∑j=1n1nμi2j2−μ¯j22=1n⋅∑j=1nμi1j2−μ¯j2⋅μi2j2−μ¯j2/1n2⋅∑j=1nμi1j2−μ¯j22⋅∑j=1nμi2j2−μ¯j22=rμPi1,Pi2.
Therefore, (T5.4) is valid, which completes the proof.

Theorem 6.

The nonmembership component rνwPi1,Pi2 in the weighted PF correlation coefficient γwPi1,Pi2 for two PF characteristics Pi1 and Pi2 satisfies the following properties:

  • (T6.1) rνwPi1,Pi2=rνwPi2,Pi1;

  • (T6.2) rνwPi1,Pi2=1 if νi1j=νi2j for all cj ∈ C;

  • (T6.3) |rνwPi1,Pi2|≤1;

  • (T6.4) rνwPi1,Pi2=rνPi1,Pi2 if w = (1/n, 1/n, ⋯, 1/n).

Proof.

The proofs of this theorem are analogous to those of Theorem 5.

Theorem 7.

The indeterminacy component rπwPi1,Pi2 in the weighted PF correlation coefficient γwPi1,Pi2 for two PF characteristics Pi1 and Pi2 satisfies the following properties:

  • (T7.1) rπwPi1,Pi2=rπwPi2,Pi1;

  • (T7.2) rπwPi1,Pi2=1 if πi1j=πi2j for all cj∈C;

  • (T7.3) |rπwPi1,Pi2|≤1;

  • (T7.4) rπwPi1,Pi2=rπPi1,Pi2 if w = (1/n, 1/n, ⋯, 1/n).

Proof.

The proofs of this theorem are analogous to those of Theorem 5.

Theorem 8.

The weighted PF correlation coefficient γwPi1,Pi2 between two PF characteristics Pi1 and Pi2 satisfies the following properties:

  • (T8.1) γwPi1,Pi2=γwPi2,Pi1;

  • (T8.2) γwPi1,Pi2=1 if Pi1=Pi2;

  • (T8.3) |γwPi1,Pi2|≤1;

  • (T8.4) γwPi1,Pi2=γPi1,Pi2 if w = (1/n, 1/n, ⋯, 1/n).

Proof.

(T8.1) is straightforward from (T5.1), (T6.1), and (T7.1). For (T8.2), the assumption Pi1=Pi2 indicates that μi1j=μi2j, νi1j=νi2j, and πi1j=πi2j hold for all cj∈C. According to (T5.2), (T6.2), and (T7.2), it is known that rμwPi1,Pi2=rνwPi1,Pi2=rπwPi1,Pi2=1, which leads to γwPi1,Pi2=1. Namely, (T8.2) is valid. For (T8.3), by means of (T5.3), (T6.3), and (T7.3), it can be concluded that |γwPi1,Pi2|≤1, i.e., (T8.3) is valid. For (T8.4), when the weight vector w = (1/n, 1/n, ⋯, 1/n), the weighted PF correlation coefficient γwPi1,Pi2 will be reduced to the PF correlation coefficient γPi1,Pi2 using Definition 5 and the properties in (T5.4), (T6.4), and (T7.4). This completes the proof.

5. PROPOSED PF TOPSIS METHOD

This section attempts to establish a novel PF TOPSIS method for addressing MCDA problems under complex PF uncertainty and identifying the ultimate priority orders of all candidate alternatives. In contrast to the distance measures commonly used in the existing TOPSIS techniques, this paper constructs new closeness measures using the proposed (weighted) PF correlation coefficients and further develops novel PF correlation-based closeness index. Figure 1 demonstrates relevant theoretical concepts of the PF TOPSIS methodology.

Figure 1

Theoretical bases of the proposed methodology.

The proposed methodology is based on the principle that the best compromise alternative should have the highest positive relationship with the positive-ideal PF solution and the lowest negative relationship with the negative-ideal PF solution. To put the assertion more concretely, the developed approach starts with the determination of new closeness measures.

Definition 6.

Let Pi, P+, and P− be the PF characteristics of an alternative zi ∈ Z, the positive-ideal PF solution z+, and the negative-ideal PF solution z−, respectively. Let MI (Pi) and MII (Pi) denote the Type I and Type II closeness measures, respectively, based on the PF correlation coefficient γ for alternative zi; they are defined as follows:

MIPi=1−γPi,P−2−γPi,P+−γPi,P−,
MIIPi=1+γPi,P+2+γPi,P++γPi,P−.
Consider the relationships among the PF characteristics Pi, P+, and P− based on the PF correlation coefficient γ in Definition 4. As a whole, a positive γ (Pi, P+) (or γ (Pi, P−)) represents that there is a positive association between Pi and P+ (or P−), whereas a negative γ (Pi, P+) (or γ (Pi, P−)) indicates a negative association. More specifically, the larger the γ (Pi, P+) is, the better the PF characteristic Pi is. In contrast, the larger the γ (Pi, P−) is, the worse the PF characteristic Pi is. The Type I closeness measure MI (Pi) is an attempt to concretize the abovementioned choice rationale. On the other side, the smaller the γ (Pi, P+) is, the worse the PF characteristic Pi is; conversely, the smaller the γ (Pi, P−) is, the better the PF characteristic Pi is. The Type II closeness measure MII (Pi) is an attempt to concretize the second choice rationale.

Theorem 9.

For each PF characteristic Pi in the PF decision matrix p, the Type I closeness measure MI (Pi) satisfies the following properties:

  • (T9.1) 0 ≤ MI (Pi) ≤ 1;

  • (T9.2) MI (Pi) = 1 if γ (Pi, P+) = 1;

  • (T9.3) MI (Pi) = 0 if γ (Pi, P−) = 1;

  • (T9.4) MI (P−) = 0 and MI (P+) - 1.

Proof.

For (T9.1), it is known that −1 ≤ γ (Pi, P+) ≤ 1 and −1 ≤ γ (Pi, P#) ≤ 1 based on the property in (T4.3). It follows that 0 ≤ 1 − γ (Pi, P+) ≤ 2, 0 ≤ 1 − γ (Pi, P−) ≤ 2, and 0 ≤ 2 − γ (Pi, P+) − γ Pi, P− ≤ 4. Thus, the property 0 ≤ MI (Pi) ≤ 1 can be readily inferred. For (T9.2), it is easily seen that the assumption γ (Pi, P+) = 1 results in MI (Pi) = (1 − γ (Pi, P−)) / 2 − 1 − γ (Pi, P−)) = 1. For (T9.3), when γ (Pi, P−) = 1, it can be obtained that MI (Pi) = (1 − 1) / (2 − γ (Pi, P+) − 1) = 0. For (T9.4), one has γ (P+, P+) = 1 and γ (P−, P−) = 1 according to (T4.2). Next, the properties MI (P+) = 1 and MI (P−) = 0 can be evidently inferred from (T9.2) and (T9.3), respectively. This establishes the theorem.

Furthermore, let us examine the property in (T9.2). Notice that there is a perfect positive correlation between Pi and P+ because γ (Pi, P+) = 1. The stronger the positive relationship between Pi and P+, the better the alternative zi is. It is easy to see that the Type I closeness measure in Definition 6 generates an acceptable and desirable result, that is, MI (Pi) = 1. Consider the property in (T9.3). One can also observe that there is a perfect positive correlation between Pi and P− because γ (Pi, P−) = 1. The stronger the positive relationship between Pi and P−, the worse the alternative zi is. It is clear that the Type I closeness measure in Definition 6 can produce a reasonable result, that is, MI (Pi) = 0.

Theorem 10.

For each PF characteristic Pi in the PF decision matrix p, the Type II closeness measure MII (Pi) satisfies the following properties:

  • (T10.1) 0 ≤ MII (Pi) ≤ 1;

  • (T10.2) MII (Pi) = 0 if γ (Pi, P+) = −1;

  • (T10.3) MII (Pi) = 1 if γ (Pi, P−) = −1;

  • (T10.4) MII (P−) ≤ MII (P+).

Proof.

For (T10.1), based on −1 ≤ γ (Pi, P+) ≤ 1 and −1 ≤ γ (Pi, P− ≤ 1 from (T4.3), it can be easily shown that 0 ≤ 1 + γ (Pi, P+) ≤ 2, 0 ≤ 1 + γ (Pi, P−) ≤ 2. It follows that 0 ≤ 2 + γ (Pi, P+) + γ (Pi, P−) ≤ 4. Thus, the property 0 ≤ MII (Pi) ≤ 1 can be easily acquired. For (T10.2), the assumption γ (Pi, P+) −1 indicates that MII (Pi) = (1 − 1) / (2 − 1 + γ (Pi, P−)) = 0. For (T10.3), the assumption γ (Pi, P−) = −1 leads to MII (Pi) = (1 + γ (Pi, P+)) / (2 + γ (Pi, P−) − 1) = 1. For (T10.4), based on the properties of (T4.1) and (T4.2), it is known that γ (P+, P−) = γ (P−, P+) and γ (P−, P−) = γ (P+, P+) = 1. By use of Definition 6, the following results can be obtained:

MIIP−=1+γP−,P+2+γP−,P++γP−,P−=1+γP−,P+3+γP−,P+,
MIIP+=1+γP+,P+2+γP+,P++γP+,P−=23+γP+,P−=23+γP−,P+.
Because γ (P−, P+) ≤ 1 according to (T4.3), it can be concluded that MII (P−) ≤ MII (P+), which establishes the theorem.

Again, let us explore the property in (T10.2). It is worth noting that there is a perfect negative correlation between Pi and P+ in case of γ (Pi, P+) = −1. The stronger the negative relationship between Pi and P+, the worse the alternative zi is. It can be observed that the Type II closeness measure in Definition 6 can yield a reasonable result, that is, MII (Pi) = 0. Next, consider the property in (T10.3). There is a perfect negative correlation between Pi and P− because γ (Pi, P−). The stronger the negative relationship between Pi and P−, the better the alternative zi is. Thus, it is clearly known that the Type II closeness measure in Definition 6 generates an acceptable result, that is, MII (Pi) = 1.

It is worth stressing that both MI (Pi) and MII (Pi) are appropriate for the specialized situations. To acquire a balanced consequence in the proposed PF TOPSIS procedure, this paper provides a comprehensive measure named as the PF correlation-based closeness index to simultaneously take into account both the Type I and Type II closeness measures.

Definition 7.

Let MI (Pi) and MII (Pi) be the Type I and Type II closeness measures, respectively, for alternative zi ∈ Z. Let ξ denote a closeness parameter, where 0 ≤ ξ ≤ 1. The PF correlation-based closeness index I(Pi) of alternative zi is defined as follows:

IPi=ξ⋅MIPi+1−ξ⋅MIIPi.
The closeness parameter ξ is referred to as the influence of the Type I and Type II closeness measures. It is supposed to vary (over 0 ≤ ξ ≤ 1) according to the emphasis on either MI (Pi), or MII (Pi), or both. The larger ξ value indicates that the specification of the PF correlation-based closeness index I (Pi) would focus on the Type I closeness measure MI (Pi). The smaller ξ value means that the specification of the I (Pi) focuses on the Type II closeness measure MII (Pi). As is apparent, the extreme case ξ = 1 produces the pure MI (Pi) results, while the extreme case ξ = 0 yields the pure MII (Pi) results.

Theorem 11.

For each PF characteristic Pi in the PF decision matrix p, the PF correlation-based closeness index I (Pi) satisfies the following properties:

  • (T11.1) 0 ≤ I (Pi) ≤ 1;

  • (T11.2) I (Pi) = MI (Pi) if ξ = 1;

  • (T11.3) I (Pi) = MII (Pi) if ξ = 0;

  • (T11.4) I (P−) ≤ I (P+).

Proof.

(T11.1) is evident according to Definition 7 and the properties in (T9.1) and (T10.1). (T11.2) and (T11.3) are straightforward from Definition 7. For (T11.4), it is known that MI (P−) ≤ MI (P+) because MI (P−) = 0 and MI (P+) using the property in (T9.4). Moreover, one has MII (P−) ≤ MII (P+) from (T10.4). It directly follows that I (P−) ≤ I (P+). This completes the proof.

Next, consider the weighted situation in which the weight vector w = (w1, w2, ⋯, wn) is incorporated into the definitions of closeness measures.

Definition 8.

Let Pi, P+, and P− be the PF characteristics of an alternative zi ∈ Z, the positive-ideal PF solution z+, and the negative-ideal PF solution z−, respectively. Let wj be the weight of criterion cj ∈ C with wj ∈ [0, 1] and ∑j=1nwj=1. Let MIwPi and MIIwPi denote the weighted Type I and Type II closeness measures, respectively, based on the PF correlation coefficient γ for alternative zi; they are defined as follows:

MIwPi=1−γwPi,P−2−γwPi,P+−γwPi,P−,
MIIwPi=1+γwPi,P+2+γwPi,P++γwPi,P−.

Theorem 12.

For each PF characteristic Pi in the PF decision matrix p, the weighted Type I closeness measure MIwPi satisfies the following properties:

  • (T12.1) 0≤MIwPi≤1;

  • (T12.2) MIwPi=1 if γwPi,P+=1;

  • (T12.3) MIwPi=0 if γwPi,P−=1;

  • (T12.4) MIwP−=0 and MIwP+=1;

  • (T12.5) MIwPi=MIPi if w = (1/n, 1/n, ⋯, 1/n).

Proof.

For (T12.1), by use of (T8.3), it is known that −1 ≤ γw (Pi, P+) ≤ 1 and −1 ≤ γw (Pi, P#). Thus, it is easily observed that 0 ≤ 1 − γw (Pi, P+) ≤ 2, 0 ≤ 1− γw (Pi, P−) ≤ 2, and 0 ≤ 2 − γw (Pi, P+ − γw (Pi, P−) ≤ 4. As a result, the property 0≤MIwPi≤1 is valid. For (T12.2), when γw (Pi, P+) = 1, one has MIwPi=1−γwPi,P−/2−1−γwPi,P−=1. For (T12.3), when γw (Pi, P−) = 1, one obtains MIwPi=1−1/2−γwPi,P+−1=0. For (T12.4), because γw (P+, P+) = 1 and γw (P−, P−) = 1 based on (T8.2), the properties MIwP+=1 and MIwP−=0 can be acquired from (T12.2) and (T12.3), respectively. For (T12.5), when w = (1/n, 1/n, ⋯, 1/n, it is known that γw (Pi, P+) = γ (Pi, P+) and γw (Pi, P−) = γ (Pi, P−) based on (T8.4). Therefore, the result MIwPi=MIPi is valid. This completes the proof.

Theorem 13.

For each PF characteristic Pi in the PF decision matrix p, the weighted Type II closeness measure MII (Pi) satisfies the following properties:

  • (T13.1) 0≤MIIwPi≤1;

  • (T13.2) MIIwPi=0 if γwPi,P+=−1;

  • (T13.3) MIIwPi=1 if γwPi,P−=−1;

  • (T13.4) MIIwP−≤MIIwP+;

  • (T13.5) MIIwPi=MIIPi if w = (1/n, 1/n, ⋯, 1/n).

Proof.

For (T13.1), based on −1 ≤ γw (Pi, P+) ≤ 1 and −1 ≤ γw (Pi, P−) ≤ 1 from (T8.3), it is evident that 0 ≤ 1 + γw (Pi, P+) ≤ 2, 0 ≤ 1 + γw (Pi, P−) ≤ 2, and 0 ≤ 2 + γw (Pi, P+) + γw (Pi, P−) ≤ 4. It follows that the property 0≤MIIwPi≤1 is valid. For (T13.2), the assumption γw (Pi, P+) = −1 results in MIIwPi=1−1/2−1+γwPi,P−=0. For (T13.3), the assumption γw (Pi, P−) = −1 leads to MIIwPi=1+γwPi,P+/2+γwPi,P+−1=1. For (T13.4), based on the properties of (T8.1) and (T8.2), it is known that γw (P+, P−) = γw (P−, P+) and γw (P−, P− = γw (P+, P+ = 1. According to Definition 8, the following results can be obtained:

MIIwP−=1+γwP−,P+2+γwP−,P++γwP−,P−=1+γwP−,P+3+γwP−,P+,
MIIwP+=1+γwP+,P+2+γwP+,P++γwP+,P−=23+γwP+,P−=23+γwP−,P+.
Because γw (P−, P+) ≤ 1 according to (T8.3), it can be concluded that MIIwP−≤MIIwP+. For (T13.5), when w = (1/n, 1/n, ⋯, 1/n, it is known that γw (Pi, P+) = γP (i, P+ and γw (Pi, P−) = γ (Pi, P−) based on (T8.4). It follows that MIIwPi=MIIPi. This establishes the theorem.

Definition 9.

Let MIwPi and MIIwPi be the weighted Type I and Type II closeness measures, respectively, for alternative zi ∈ Z. Let ξ denote a closeness parameter, where 0 ≤ ξ ≤ 1. The weighted PF correlation-based closeness index Iw (Pi) of alternative zi is defined as follows:

IwPi=ξ⋅MIwPi+1−ξ⋅MIIwPi.

Theorem 14.

For each PF characteristic Pi in the PF decision matrix p, the weighted PF correlation-based closeness index Iw (Pi) satisfies the following properties:

  • (T14.1) 0 ≤ Iw (Pi) ≤ 1;

  • (T14.2) IwPi=MIwPi if ξ = 1;

  • (T14.3) IwPi=MIIwPi if ξ = 0;

  • (T14.4) Iw (P−) ≤ Iw (P+);

  • (T14.5) Iw (Pi) = I (>Pi) if w = (1/n, 1/n, ⋯, 1/n).

Proof.

(T14.1) is evident according to Definition 9 and the properties in (T12.1) and (T13.1). (T14.2) and (T14.3) are straightforward from Definition 9. For (T14.4), it is known that MIwP−≤MIwP+ because MIwP−=0 and MIwP+=1 using the property in (T12.4). Additionally, one has MIIwP−≤MIIwP+ from (T13.4). Thus, one can obtain Iw (P−) ≤ Iw (P+). For (T14.5), the result of Iw (Pi) = I (Pi) can be inferred based on the properties in (T12.5) and (T13.5). This completes the proof.

Based on the proposed concepts of PF correlation coefficients, Type I and Type II closeness measures, and PF correlation-based closeness indices, this paper proposes a novel PF TOPSIS method for addressing MCDA problems involving PF information. Figure 2 illustrates the implementation procedure of the PF TOPSIS method.

Figure 2

Algorithmic procedure of the proposed methodology.

The algorithmic procedure of the proposed PF TOPSIS methodology can be summarized as follows:

  • Step 1: Formulate an MCDA problem with the set of candidate alternatives Z = {z1, z2, ⋯, zm} and the set of evaluative criteria C = {c1, c2, ⋯, cn}, which is divided into CB and CC.

  • Step 2: Designate the weight vector w = (w1, w2, ⋯, wn for the n evaluative criteria. Set w = (1/n, 1/n, ⋯, 1/n in the unweighted situation.

  • Step 3: Establish the PF evaluative rating pij of each alternative zi ∈ Z with respect to criterion cj ∈ C.

  • Step 4: Form the PF decision matrix p(=[pij]m×n) using Equation (10). Define the PF characteristic Pi of each zi ∈ Z using Equation (11).

  • Step 5: Identify the characteristics P+ of the positive-ideal PF solution z+ and P− of the negative-ideal PF solution z− using Equations (14) and (15), respectively.

  • Step 6: Compute the membership components (i.e., rμwPi,P+ and rμwPi,P−, the nonmembership components (i.e., rνwPi,P+ and rνwPi,P−), and the indeterminacy components (i.e., rπwPi,P+ and rπwPi,P−) using Equations (21–23), respectively, for each zi ∈ Z.

  • Step 7: Apply Equation (20) to derive the weighted PF correlation coefficients γw (Pi, P+) and γw (Pi, P−) between Pi and P+ and between Pi and P−, respectively, for each zi ∈ Z.

  • Step 8: Employ Equations (27) and (28) to compute the weighted Type I closeness measure MIwPi and the weighted Type II closeness measure MIIwPi, respectively, for each zi ∈ Z.

  • Step 9: Set the closeness parameter ξ, where 0 ≤ ξ ≤ 1. Calculate the weighted PF correlation-based closeness index Iw (Pi) using Equation (29) for each zi ∈ Z.

  • Step 10: Determine the ultimate priority orders among the m alternatives according to the descending order of the Iw (Pi) values.

6. PRACTICAL APPLICATION

This section attempts to demonstrate an illustrative application in a practical decision-making problem of hospitalization rehabilitation treatments for stroke patients with the purpose of validating the feasibility and validity of the proposed PF TOPSIS methodology. In the following, this section first describes the problem background of the medical decision concerning rehabilitation treatments for hospitalized stroke patients at acute stage.

Stroke is referred to as cerebrovascular accident. The primary cause of stroke is obstructed blood flow resulting in neurological deficits or brain hypoxia and ischemia. Stroke leads to the death or impairment of brain tissue, which may cause temporary or permanent dysfunction. According to the onset time, stroke can be divided into acute, post-acute (or sub-acute), and chronic stages. The acute stage occurs after the onset of acute stroke, and the post-acute stage occurs after patients are discharged from hospitalization for the acute phase. Finally, the chronic stage follows the post-acute stage. In particular, early rehabilitation treatment can benefit patients’ recovery of walking ability, enhance their independence in daily life, and reduce length of hospital stay. Therefore, rehabilitation treatment at the acute stage in the hospital is a crucial issue for stroke patients.

Linkou Chang Gung Memorial Hospital is located in Taoyuan City, Taiwan. The hospital’s total number of beds is approximately 3,700, and the service team comprises more than 9,000 people. Annually, the hospital provides service for 4 million outpatient clinic visits, 200,000 emergency visits, and 100,000 hospitalizations. It is the largest medical institution in Taiwan. This practical application case study explored stroke rehabilitation practices and challenges in the department of nursing of Linkou Chang Gung Memorial Hospital. Using multiple criteria as stipulated by the relevant authorities, the priorities for stroke rehabilitation treatments at the acute stage were evaluated, and the findings were used as references for clinical guidelines.

The department of nursing proposed five hospitalization rehabilitation treatments (consisting of turning over (z1), positioning (z2), passive range of motion (z3), music rehabilitation exercise (z4), and air bed (z5)) and eight evaluative criteria (consisting of pressure sore incidence (c1), aspiration pneumonia incidence (c2), arthrogryposis incidence (c3), shoulder subluxation incidence (c4), length of hospital stay (c5), degree of disability (c6), functional abilities for daily life (c7), and medical satisfaction (c8)) to assess these alternatives according to patients’ conditions. Because of the particularity of inpatient stroke rehabilitation, the medical decision-making problem of hospitalization rehabilitation treatments becomes a very complicated and ambiguous MCDA problem. To validate the effectiveness and practicability of the PF TOPSIS method, this paper employs the developed approach and techniques to assist the priority ranking of rehabilitation care services for hospitalized patients with stroke and cerebrovascular diseases.

In Step 1, the MCDA problem under study is defined by five hospitalization rehabilitation treatments and eight criteria for evaluating the alternatives. The set of candidate alternatives is denoted by Z = {z1, z2, ⋯, z5}, and the set of evaluative criteria is denoted by C = {c1, c2, ⋯, c8}, in which CB = {c1, c2, ⋯,c6} and CC = {c7, c8}.

In Step 2, based on the authority’s knowledge and expertise in the Department of Nursing at Linkou CGMH, the weight vector for the eight evaluative criteria was designated as follows: w = (0.12, 0.15, 0.17, 0.10, 0.08, 0.17, 0.14, 0.07). In Step 3, the PF evaluative rating pij of each alternative zi ∈ Z with respect to criterion cj ∈ C were established by the authority, as indicated in Table 1.

zi c1 c2 c3 c4 c5 c6 c7 c8
z1 (0.05, 0.92) (0.74, 0.37) (0.51, 0.46) (0.63, 0.57) (0.38, 0.81) (0.99, 0.00) (0.13, 0.93) (0.38, 0.68)
z2 (0.12, 0.88) (0.13, 0.87) (0.25, 0.76) (0.49, 0.64) (0.76, 0.39) (0.63, 0.60) (0.38, 0.68) (0.37, 0.85)
z3 (0.74, 0.30) (0.98, 0.00) (0.13, 0.87) (0.37, 0.73) (0.26, 0.71) (0.48, 0.56) (0.52, 0.51) (0.53, 0.55)
z4 (0.87, 0.24) (0.99, 0.00) (0.12, 0.82) (0.36, 0.66) (0.24, 0.89) (0.49, 0.57) (0.51, 0.71) (0.65, 0.32)
z5 (0.01, 0.97) (0.98, 0.00) (0.86, 0.16) (0.99, 0.01) (0.35, 0.84) (0.85, 0.11) (0.13, 0.86) (0.88, 0.21)
Table 1

Data of the PF evaluative ratings.

In Step 4, the PF decision matrix was constructed based on the PF evaluative ratings in Table 1, that is, p=[pij]5×8=[(μij,vij)]5×8. The PF characteristic Pi of each zi was identified Pi={〈c1,pi1〉,〈c2,pi2〉,⋯,〈c8,pi8〉}={〈c1,(μi1,νi1)〉,〈c2,(μi2,νi2)〉,⋯,〈c8,(μi8,νi8)〉}.

In Step 5, the characteristics P+ and P− of the positive- and negative-ideal PF solutions z+ and z−, respectively, were obtained as follows:

P+=〈c1,p+1〉,〈c2,p+2〉,⋯,〈c8,p+8〉=〈c1,(0.01,0.97)〉,〈c2,(0.13,0.87)〉,〈c3,(0.12,0.87)〉,〈c4,(0.36,0.73)〉,〈c5,(0.24,0.89)〉,〈c6,(0.48,0.60)〉,〈c7,(0.52,0.51)〉,〈c8,(0.88,0.21)〉,
P−=〈c1,p−1〉,〈c2,p−2〉,⋯,〈c8,p−8〉=〈c1,(0.87,0.24)〉,〈c2,(0.99,0.00)〉,〈c3,(0.86,0.16)〉,〈c4,(0.99,0.01)〉,〈c5,(0.76,0.39)〉,〈c6,(0.99,0.00)〉,〈c7,(0.13,0.93)〉,〈c8,(0.37,0.85)〉.
In Step 6, the membership, nonmembership, and indeterminacy components of the weighted PF correlation coefficients between Pi and P+ and between Pi and P− were derived for each zi ∈ Z. The computation results of rμwPi,P+, rνwPi,P+, rπwPi,P+, rμwPi,P−, rνwPi,P−, and rπwPi,P− are revealed in Table 2. In Step 7, based on the obtained components, the determination results of γw (Pi, P+) and γw (>Pi, P−) for each zi are presented in Table 2 as well. In Step 8, the weighted Type I closeness measures for each zi were calculated as follows: MIwP1=0.3664, MIwP2=0.6109, MIwP3=0.6403, MIwP4=0.6262, and MIwP5=0.4425. The weighted Type II closeness measures were computed as follows: MIIwP1=0.3665, MIIwP2=0.6263, MIIwP3=0.6461, MIIwP4=0.6373, and MIIwP5=0.4732.

zi rμwPi,P+ rνwPi,P+ rπwPi,P+ γw (Pi, P+) rμwPi,P− rνwPi,P− rπwPi,P− γw (Pi, P−)
z1 −0.4106 0.0553 −0.4457 −0.2670 0.4694 0.1351 0.1971 0.2672
z2 0.6241 0.7283 0.5484 0.6336 −0.3628 −0.1924 −0.3491 −0.3015
z3 −0.2604 0.2067 0.3604 0.1022 0.0300 −0.3781 −0.5710 −0.3064
z4 −0.2237 0.1228 0.2256 0.0416 0.1275 −0.3553 −0.6878 −0.3052
z5 −0.4605 −0.1008 0.0895 −0.1572 0.6956 0.0629 0.5537 0.4374

PF, Pythagorean fuzzy.

Table 2

Results of the weighted PF correlation coefficients.

In Step 9, the closeness parameter was designated as follows: ξ = 0.5. Next, the weighted PF correlation-based closeness indices were acquired as follows: Iw (P1 = 0.3665, Iw (P2) = 0.6186, Iw (P3) = 0.6432, Iw (P4) = 0.6318, and Iw (P5) = 0.4578. In Step 10, based on the descending order of these Iw (Pi) values, the ultimate priority ranking z3 ≻ z4 ≻ z2 ≻ z5 ≻ z1 was obtained as a useful decision-aiding suggestion for the MCDA problem of hospitalization rehabilitation treatments.

Furthermore, more comparative discussions of the determination results are conducted to examine the effectiveness and reasonability of the practical application. Concerning the rehabilitation treatment case, some comprehensive comparisons are depicted in Figures 3 and 4. In regard to each zi ∈ Z, the contrasts of the weighted Type I closeness measure MIwPi, the weighted Type II closeness measure MIIwPi, and the weighted PF correlation-based closeness index Iw (Pi) are presented in Figure 3. The obtained MIwPi and MIIwPi are very close with respect to each alternative, especially for the alternatives z2, z3, and z4. More precisely, the absolute differences between MIwPi and MIIwPi for each alternative were acquired as follows: 0.0001 for z1, 0.0154 for z2, 0.0058 for z3, 0.0111 for z4, and 0.0307 for z5. The distribution patterns of MIwPi and MIIwPi are very similar concerning all of the five alternatives. Accordingly, the general pattern of Iw (Pi) across all alternatives is concordant with the respective findings based on MIwPi and MIIwPi, as demonstrated in Figure 3.

Figure 3

Comparison of closeness measures and PF correlation-based closeness indices.

Based on the descending orders of the MIwPi, MIIwPi, and Iw (Pi) values, three priority ranking orders of alternatives are contrasted in Figure 4. The consistent results can be found in terms of the priority orders of the five candidate alternatives. More specifically, the priority ranking is given as z3 ≻ z4 ≻ z2 ≻ z5 ≻ z1 regardless of the comparison bases of MIwPi, MIIwPi, and Iw (Pi). Because the priority ranking orders are in line with each other, the stability of the results is all the more impressive.

Figure 4

Contrast of the priority ranking orders of alternatives via MIwPi, MIIwPi, amd Iw (Pi).

7. COMPARATIVE STUDIES

This section conducts some comparative analyses with previous researches to validate the effectiveness of the proposed approach and highlight the merits of the study.

It has to be stressed that the proposed PF TOPSIS method differs considerably from the existing TOPSIS techniques by its identification of relative closeness. As opposed to the current distance-based closeness indices, the proposed methodology employs novel concepts of PF correlations and two types of closeness measures to present the correlation-based closeness index, which is significantly different from the previous studies. Therefore, the comparative studies focus on the contrast of the obtained TOPSIS solutions based on correlation-based and distance-based closeness indices.

The comparative analyses investigate four TOPSIS techniques, consisting of the PF TOPSIS and traditional TOPSIS methods based on weighted evaluative ratings or weighted distances. As mentioned before, the current PF TOPSIS techniques presented by Gul and Ak [34], Liang and Xu [33], Liang et al. [35], Zeng et al. [32], Zhang and Xu [22], and Zhou et al. [36] employed PF distances as a separation measure to define relevant concepts of closeness indices. Referring the common TOPSIS structure based on distance-based closeness indices in these researches, this paper provides two PF TOPSIS techniques using different weighted approaches (i.e., weighted evaluative ratings and weighted distances). On the other hand, to examine the contributions and the advantages of the proposed methodology relative to the traditional TOPSIS methods, this paper converts PF evaluative ratings into crisp data via the normalized score functions. Analogously, this paper presents two traditional TOPSIS techniques based on weighted evaluative ratings and weighted distances.

The first comparative approach is the PF TOPSIS technique based on weighted evaluative ratings. For each pij (= (μij, νij)) in the PF decision matrix p, the weighted PF evaluative rating pijw=μijw,νijw is computed as follows:

pijw=wj⋅pij=1−1−μij2wj,νijwj.
The weighted PF characteristic Piw is given by:
Piw=〈c1,pi1w〉,〈c2,pi2w〉,⋯,〈cn,pinw〉.
Let z+w and z−w denote the weighted positive- and negative-ideal PF solutions, respectively. The PF characteristics of z+w and z−w are represented as follows:
P+w=〈c1,p+1w〉,〈c2,p+2w〉,⋯,〈cn,p+nw〉,
P−w=〈c1,p−1w〉,〈c2,p−2w〉,⋯,〈cn,p−nw〉,
where the weighted positive- and negative-ideal PF values within P+w=μ+jw,ν+jw and P−w=μ−jw,ν−jw are defined as follows:
p+jw=∨i=1mpijw=maxi=1mμijw,mini=1mνijwif  cj∈CB,∧i=1mpijw=mini=1mμijw,maxi=1mνijwif  cj∈CC,
p−jw=∧i=1mpijw=mini=1mμijw,maxi=1mνijwif  cj∈CB,∨i=1mpijw=maxi=1mμijw,mini=1mνijwif  cj∈CC.
Based on Zhang and Xu’s PF distance measure [22], the normalized distances between Piw and P+w as well as between Piw and P−w are as follows:
DPiw,P+w=12n⋅∑j=1n|μijw2−μ+jw2|+|νijw2−ν+jw2|+|πijw2−π+jw2|,
DPiw,P−w=12n⋅∑j=1n|μijw2−μ−jw2|+|νijw2−ν−jw2|+|πijw2−π−jw2|.
Employing the first comparative approach, the closeness coefficient CCPiw of alternative zi is defined as follows:
CCPiw=DPiw,P−wDPiw,P+w+DPiw,P−w.
The second comparative approach is the PF TOPSIS technique based on weighted distances. The normalized weighted PF distances between Pi and P+ as well as between Pi and P− are calculated as follows:
DwPi,P+=12n⋅∑j=1nwj⋅|μij2−μ+j2|+|νij2−ν+j2|+|πij2−π+j2|,
DwPi,P−=12n⋅∑j=1nwj⋅|μij2−μ−j2|+|νij2−ν−j2|+|πij2−π−j2|.
Employing the second comparative approach, the closeness coefficient CCw (Pi) of alternative zi is defined as follows:
CCwPi=DwPi,P−DwPi,P++DwPi,P−.
The third comparative approach is the traditional TOPSIS technique based on weighted evaluative ratings. Zhang and Xu’s developed a score function [22] for PF values and applied it to compare the magnitudes of PF values. By modifying their definition, this paper converts the PF data into crisp numbers using the normalized outcomes of score functions. Let p¯ijw denote the weighted evaluative rating of an alternative zi ∈ Z with respect to criterion cj ∈ C. It is noted that p¯ijw is expressed as a crisp number and can be determined by use of the normalized score function. For each pijw, the weighted evaluative rating p¯ijw is acquired by the normalized score function as follows:
p¯ijw=1+μijw2−νijw22=2−1−μij2wj−νij2wj2.
The weighted characteristic P¯iw is given by:
P¯iw=〈c1,p¯i1w〉,〈c2,p¯i2w〉,⋯,〈cn,p¯inw〉.
The characteristics of the weighted positive- and negative-ideal solutions z¯+w and z¯−w are represented as follows:
P¯+w=〈c1,p¯+1w〉,〈c2,p¯+2w〉,⋯,〈cn,p¯+nw〉,
P¯−w=〈c1,p¯−1w〉,〈c2,p¯−2w〉,⋯,〈cn,p¯−nw〉,
where the weighted positive- and negative-ideal values within P¯+w and P¯−w are defined as follows:
p¯+jw=maxi=1mp¯ijwif  cj∈CB,mini=1mp¯ijwif  cj∈CC,
p¯−jw=mini=1mp¯ijwif  cj∈CB,maxi=1mp¯ijwif  cj∈CC.
Based on the Hamming distance model, the closeness coefficient CCP¯iw of alternative zi is defined as follows:
CCP¯iw=∑j=1n|p¯ijw−p¯−jw|∑j=1n|p¯ijw−p¯+jw|+|p¯ijw−p¯−jw|.
The fourth comparative approach is the traditional TOPSIS technique based on weighted distances. Let p¯ij denote the evaluative rating of an alternative zi ∈ Z with respect to criterion cj ∈ C; it is defined as follows:
p¯ij=1+μij2−νij22.
The characteristic P¯i is given by:
P¯i=〈c1,p¯i1〉,〈c2,p¯i2〉,⋯,〈cn,p¯in〉.
The characteristics of the positive- and negative-ideal solutions z¯+ and z¯− are represented as follows:
P¯+=〈c1,p¯+1〉,〈c2,p¯+2〉,⋯,〈cn,p¯+n〉,
P¯−=〈c1,p¯−1〉,〈c2,p¯−2〉,⋯,〈cn,p¯−n〉,
where the positive- and negative-ideal values within P¯+ and P¯− are defined as follows:
p¯+j=maxi=1mp¯ijif  cj∈CB,mini=1mp¯ijif  cj∈CC,
p¯−j=mini=1mp¯ijif  cj∈CB,maxi=1mp¯ijif  cj∈CC.
Based on the weighted Hamming distance model, the closeness coefficient CCwP¯i of alternative zi is defined as follows:
CCwP¯i=∑j=1nwj⋅|p¯ij−p¯−j|∑j=1nwj⋅|p¯ij−p¯+j|+|p¯ij−p¯−j|.
Consider the rehabilitation treatment case. This paper conducts the comparison of the application results rendered by the proposed methodology and the four comparative approaches within the PF uncertain and certain environments. Table 3 indicates the obtained results, consisting of the closeness coefficients CCPiw and CCw (Pi) yielded by the PF TOPSIS methods based on weighted evaluative ratings and based on weighted distances, respectively, the closeness coefficients CCP¯iw and CCwP¯i produced by the traditional TOPSIS methods based on weighted evaluative ratings and based on weighted distances, respectively, and the weighted PF correlation-based closeness index Iw (Pi) generated by the proposed PF TOPSIS methodology. It is noted that the obtained CCPiw, CCw (Pi), and Iw (Pi result from PF uncertain information, while CCP¯iw and CCwP¯i result from certain information.

zi CCPiw CCw (Pi) CCP¯iw CCwP¯i Iw (Pi)
z1 0.5380 0.4217 0.5463 0.4286 0.3665
z2 0.8600 0.7259 0.8712 0.7480 0.6186
z3 0.6156 0.6252 0.6345 0.6231 0.6432
z4 0.5898 0.5858 0.6033 0.5911 0.6318
z5 0.3164 0.3411 0.3287 0.3288 0.4578
Table 3

Comparison of the obtained results.

Furthermore, a comprehensive contrast on the features and core concepts possessed by the five comparative methods is demonstrated in Table 4. The first and second comparative approaches belong to the distance-based PF TOPSIS model; moreover, the third and fourth comparative approaches belong to the distance-based traditional TOPSIS model. In contrast, the proposed methodology belongs to the correlation-based PF TOPSIS model.

Distance-Based PF TOPSIS Distance-Based traditional TOPSIS Correlation-Based PF TOPSIS
Comparative approach
PF TOPSIS based on weighted evaluative ratings PF TOPSIS based on weighted distances Traditional TOPSIS based on weighted evaluative ratings Traditional TOPSIS based on weighted distances Proposed PF TOPSIS methodology
Decision environment
PF context PF context Certain context Certain context PF context
Separation measure
PF distance Weighted PF distance Hamming distance Weighted Hamming distance Weighted PF correlation coefficient
Key index
PF distance-based closeness index PF distance-based closeness index Distance-based closeness index Distance-based closeness index Types I & II closeness measures PF correlation-based closeness index

PF, Pythagorean fuzzy; TOPSIS, technique for order preference by similarity to ideal solutions.

Table 4

Comparison of methodological core concepts.

To provide a better view of the comparison results, this paper puts the obtained CCPiw, CCw (Pi), CCP¯iw, CCwP¯i, and Iw (Pi) using the comparative approaches into Figure 5. Additionally, the ultimate priority ranking orders of the five candidate alternatives rendered by the comparative approaches are contrasted in Figure 6. The consistent finding in the comparative analysis is that the alternatives z2, z3, and z4 are the more appropriate rehabilitation treatments for hospitalized stroke patients at acute stage. Moreover, the priority rankings produced by the four comparative approaches are the same. Namely, the priority ranking is determined as z2 ≻ z3 ≻ z4 ≻ z1 ≻ z5 regardless of the comparison bases of CCPiw, CCw (Pi), CCP¯iw, and CCwP¯i.

Figure 5

Contrast of the obtained CCPiw, CCw (Pi), CCP¯iw, CCwP¯i, and Iw (Pi) using the comparative approaches.

Figure 6

Contrast of the obtained priority ranking orders in the comparative analyses.

From Figure 6, one can find that the ranking results yielded by the two distance-based PF TOPSIS methods are identical to the results rendered by the two distance-based traditional TOPSIS methods. In contrast to the four comparative approaches mentioned above, the proposed methodology generated a different priority ranking: z3 ≻ z4 ≻ z2 ≻ z5 ≻ z1. The most prominent difference exists in the ranking orders is the alternative in first place. The alternative z2 is ranked first based on the descending order of CCPiw, CCw (Pi), CCP¯iw, and CCwP¯i, whereas the alternative z3 is top-ranked by Iw (Pi). Because the modified score function can provide a simple and convenient manner to compare the magnitudes of PF evaluative ratings, this paper investigates the detailed evaluation data of the alternatives z2 and z3 to examine the effectiveness and reasonability the obtained results.

For the positioning treatment (z2), the following results using the modified score function were obtained: p¯21=0.1200, p¯22=0.1300, p¯23=0.2425, p¯24=0.4153, p¯25=0.7128, p¯26=0.5185, p¯27=0.3410, and p¯28=0.2072. For the passive range of motion (z3), the following results were acquired: p¯31=0.7288, p¯32=0.9802, p¯33=0.1300, p¯34=0.3020, p¯35=0.2818, p¯36=0.4584, p¯37=0.5052, and p¯38=0.4892. Recall that CB = {c1, c2, ⋯, c6 and CC = {c7, c8. Because p¯21<p¯31 and p¯22<p¯32, z2 performs better than z3 with respect to the cost criteria c1 and c2. However, z3 performs better than z2 in terms of the remaining criteria. More specifically, it is clear that p¯23>p¯33, p¯24>p¯34, p¯25>p¯35, and p¯26>p¯36 with respect to the cost criteria c3, c4, c5, and c6; moreover, p¯27<p¯37 and p¯28<p¯38 about the benefit criteria c7 and c8. That is, z3 is superior to z2 in regard to the six criteria (i.e., c3–c8). In contrast, z2 is superior to z3 only on the two criteria (i.e., c1 and c2). Thus, it is obvious that the passive range of motion is more appropriate than the positioning treatment (i.e., z3 ≻ z2) for inpatient stroke rehabilitation. Therefore, the obtained result yielded by the proposed methodology is more reasonable and acceptable than those generated by the two distance-based PF TOPSIS methods and the two distance-based traditional TOPSIS methods.

Based on the application results of the inpatient stroke rehabilitation case, the developed PF TOPSIS method using PF correlation-based closeness indices has rationality and effectiveness to assist rehabilitation care services with cerebrovascular diseases. Moreover, the proposed PF TOPSIS methodology is highly appealing in dealing with complex PF uncertainty as it allows for greater flexibility in regard to the separation measures and traditional relative closeness by employing the new PF correlations. Furthermore, the data processing steps in the proposed algorithmic procedure are relatively simple and effective to avoid the loss of original PF uncertain information. More importantly, the proposed methodology is performed around the new concepts of weighted PF correlation coefficients and Types I and II closeness measures, while the uniformity of the core structure of classical TOPSIS is still maintained.

8. CONCLUSIONS

TOPSIS is one of the famous MCDA methods. This paper has focused on the extensions of TOPSIS applied in complicated decision environments based on PF sets. PF sets have the capability of handling more uncertainty, and hence, the PF theory has been utilized to assess and improve complex MCDA problems in this study. Based on certain useful concepts of correlation measures and PF correlation-based closeness indices, the proposed PF TOPSIS methodology would produce more accurate and robust results, as demonstrated in the practical application of inpatient stroke rehabilitation at acute stage. Furthermore, the application results can assist the rehabilitation care services for stroke patients and subsequent nursing care.

In the hospitalization rehabilitation case, the results yielded by the proposed PF TOPSIS methodology have been compared with the results generated by some previous studies, and partial differences have been observed between them. Based on the comparative discussions, the obtained results based on PF correlation-based closeness indices has shown a desirable degree of reasonability. More importantly, the combination of the core structure of TOPSIS with PF information can compensate for the lack of certainty, and allow the decision maker to arrive at an acceptable solution. Further, the proposed methodology can determine the priority ranking orders of alternatives and acquire the best compromise solution within the environment involving the objective complexity of MCDA problems and the uncertainty of human subjective judgments.

In future research, the PF TOPSIS method can be considered to be used in a multipurpose decision-making system for a wide range of applications. Concretely speaking, in the future work, the proposed methodology can be applied to address multiple criteria evaluation problems in various fields, such as logistics planning [37], situation assessment [38], credit risk evaluation [7], new product development [39], and so on. Another orientation for future research could extend the proposed methodology to deal with large-scale group decision-making problems [40, 41], bi-level decision-making problems [42, 43], tri-level decision-making problems [44, 45], and fuzzy multilevel decision-making problems [46] for enhancing theoretical value and merits of the study.

ACKNOWLEDGMENTS

The authors acknowledge the assistance of the respected editor and the anonymous referees for their insightful and constructive comments, which helped to improve the overall quality of the paper. The authors are grateful for grant funding support from the Taiwan Ministry of Science and Technology (MOST 105-2410-H-182-007-MY3) and Chang Gung Memorial Hospital (BMRP 574 and CMRPD2F0203) during the study completion.

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Publication Date
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TY  - JOUR
AU  - Yu-Li Lin
AU  - Lun-Hui Ho
AU  - Shu-Ling Yeh
AU  - Ting-Yu Chen
PY  - 2018
DA  - 2018/11/01
TI  - A Pythagorean Fuzzy TOPSIS Method Based on Novel Correlation Measures and Its Application to Multiple Criteria Decision Analysis of Inpatient Stroke Rehabilitation
JO  - International Journal of Computational Intelligence Systems
SP  - 410
EP  - 425
VL  - 12
IS  - 1
SN  - 1875-6883
UR  - https://doi.org/10.2991/ijcis.2018.125905657
DO  - 10.2991/ijcis.2018.125905657
ID  - Lin2018
ER  -