Abstract
The intelligent healthcare management is of great concern to mobilize the enthusiasm of individuals and groups, and effectively use limited resources to achieve maximum health improvement by AI technology. When considering the intelligent healthcare management evaluation, the primary issues involve many uncertainties. Interval-valued fuzzy soft set, depicted by membership degree with interval form, is a more resultful means for capturing uncertainty. In this paper, the comparison issue in interval-valued fuzzy soft environment is disposed of by proposing novel score function. Later, some new properties for interval-valued fuzzy soft matrix are investigated in detail. Moreover, the objective weight is calculated by CRITIC (Criteria Importance Through Inter-criteria Correlation) method. Meanwhile, the combined weight is determined by reflecting both subjective weight and the objective weight. Then, interval-valued fuzzy soft decision-making algorithm-based CoCoSo (Combined Compromise Solution) is developed. Lastly, the validity of algorithm is expounded by the healthcare management industry evaluation issue, along with their sensitivity analysis. The main characteristics of the presented algorithm are: (1) without counterintuitive phenomena; (2) no division by zero problem; (3) have strong ability to distinguish alternatives.
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1 Introduction
Intelligent healthcare management (IHM) is based on the results of health checkups, the establishment of a dedicated health record, and targeted personalized health management programs (Coyle 2010). With the development of artificial intelligence technology, the health monitoring, health assessment and health intervention of intelligent detection instruments for users are realized.
Recently, IHM has already been successfully employed in diverse fields, such as social commerce open innovation (Davies et al. 2019), blood and marrow transplant (Repaczki-Jones et al. 2019), professionalization in medicine (Gerard 2019), education (Bean et al. 2019), logistics (Pohjosenpera et al. 2019), airway management (Schnittker et al. 2019), supply chain management (Scavarda et al. 2019). Nanshan Zhong, as an academician of the Chinese academy of engineering and a famous respiratory expert, once said, “Intelligent healthcare management is the future of life technology.” The healthcare management companies are looking for business opportunities in the blue ocean of intelligent healthcare market. However, it is not enough to rely themselves on taking the intelligent healthcare market quickly. As a consequence, they should choose an amazing artificial intelligence (AI) company to join in cooperation for upgrading their technical level. As AI leaders, Toutiao, Sensetime, Megvii, Cloudwalk and Ubtech are certainly great cooperative partners to be considered. Therefore, I propose to choose the right cooperative company to solve the problem of MCDM (multi-criteria decision making). Nevertheless, the increasingly complex decision-making environment and indecisive decision makers (DMs) make it difficult to express uncertain decision information when solving the above multi-criteria decision making (MCDM) problems (Si et al. 2019).
Interval-valued fuzzy soft set (IVFSS), firstly introduced by Son (2007), has been deemed to be a more resultful means to depict uncertain information in a greater range point when compared with soft set (SS) (Molodtsov 1999). In view of such advantage (Feng et al. 2019) of the IVFSS, it is fast becoming a hot study topic, including algebraic structure (Liu et al. 2014, 2014), matrix theory (Rajarajeswari and Dhanalakshmi 2014), parameter reduction (Ma et al. 2013; Qin and Ma 2018), information measure (Jiang et al. 2013; Wang et al. 2015; Peng and Yang 2015), data filling (Qin and Ma 2019) and decision-making technologies (Qin and Ma 2018; Yang and Peng 2017; Yang et al. 2009; Feng et al. 2010; Yuan and Hu 2012; Xiao et al. 2013; Peng and Yang 2017; Tripathy et al. 2017; Peng et al. 2017; Chen and Zou 2017; Peng and Garg 2018; Sooraj and Tripathy 2018).
When disposing of MCDM issues in interval-valued fuzzy soft environment, there are four limiting factors that are most advanced, which constitute our motivations.
-
(1)
The existing decision-making methods (Qin and Ma 2018; Yang and Peng 2017; Yang et al. 2009; Feng et al. 2010; Yuan and Hu 2012; Xiao et al. 2013; Peng and Yang 2017; Tripathy et al. 2017; Peng et al. 2017; Chen and Zou 2017; Peng and Garg 2018), disposing of dealing MCDM problems, have counterintuitive phenomena (Yang et al. 2009; Yuan and Hu 2012; Xiao et al. 2013), division by zero problem (Yang and Peng 2017) and low resolution in differentiating the optimal alternative (Peng et al. 2017; Son 2007; Chen and Zou 2017; Peng and Garg 2018). It may be unaccommodated or unfitted for experts to choose best alternative(s). The CoCoSo (Combined Compromise Solution) method, initially explored by Yazdani et al. (2019), is an effective algorithm to deal the uncertain information in a logical and commonsensible way (Wen et al. 2019). Therefore, the first motivation is to deal MCDM issues by presenting novel decision making algorithm out of three deficiencies.
-
(2)
The existing score functions (Peng and Yang 2017) can accurately rank the common alternatives. However, it cannot distinguish two IVFNs in some special cases. Hence, the second motivation is to introduce a new score function, which can have strong sense of differentiation.
-
(3)
The existing interval-valued fuzzy soft weighting determine models only focus on objective weight (Yang and Peng 2017; Xiao et al. 2013; Chen and Zou 2017) or subjective weight (Tripathy et al. 2017; Sooraj and Tripathy 2018). The subjective weight is offered by DMs based on their experience, while they neglect the underlying weight information conveyed by the evaluation information. The objective weight can be obtained from the evaluation information by some innovative methods, while they fail to take the preference information of experts into account. Moreover, it may lead drastically to increasing criteria weight of that criteria when one value in decision matrix is above/below the other values for 20–30% (within the same criteria). How to combine them is an interesting topic (Peng and Yang 2017; Peng et al. 2017; Peng and Garg 2018). Hence, the third motivation is to develop the combined weight model, which can integrate both subjective weight and objective weight.
-
(4)
Some matrix operations (Liu et al. 2014; Rajarajeswari and Dhanalakshmi 2014) on IVFSSs are not abundant enough, and their properties are not enough wonderful. Hence, the fourth motivation of this article is to present some matrix operations for IVFSSs and also discuss their interesting properties in detail.
According to the above argumentation and the characteristics of IHM industry evaluation, we present a revised MCDM method in the interval-valued fuzzy soft environment. The contributions of paper are shown as follows:
-
(1)
A novel interval-valued fuzzy soft IHM industry decision making issue based on CoCoSo method is explored, which can obtain the best alternative without counterintuitive phenomena and division by zero problem, and have a strong ability to differentiate alternatives.
-
(2)
The novel score function of IVFN is presented and their interesting properties are explored in detail.
-
(3)
The combined weight model is based on CRITIC (Criteria Importance Through Inter-criteria Correlation) and the linear weighted comprehensive method, which can reflect both subjective weight and objective weight information.
-
(4)
Some new theorems and relations based on interval-valued fuzzy soft matrix are discussed.
To process our discussion, the rest part is listed in the following. In Sect. 2, some basic notions of interval-valued fuzzy set and IVFSS are primitively retrospected. In Sect. 3, the novel score function of IVFN is explored and its properties are proved. In Sect. 4, the novel matrix-based IVFSSs are defined and discussed in detail. In Sect. 5, we develop a novel interval-valued fuzzy soft-based CoCoSo with CRITIC. In Sect. 6, a case study in intelligent health management evaluation method is presented, and the sensitive analysis is shown. In Sect. 7, some examples to illustrate the effectiveness of developed method under interval-valued fuzzy soft environment are given. Section 8 makes some conclusions.
2 Preliminaries
This section primarily reviews some notions of interval-valued fuzzy set and interval-valued fuzzy soft set.
Definition 1
(Zadeh 1975) An interval-valued fuzzy set (IVFS) I in U is given by
where \(0\le I^-(\chi )\le I^+(\chi )\le 1\). For simplicity, we call \(i=[I^-(\chi ),I^+(\chi )]\) an interval-valued fuzzy number (IVFN) denoted by \(i=[i^-,i^+]\).
Definition 2
(Xu and Da 2002) Let \(\chi =[\chi ^-,\chi ^+]\) and \(y=[y^-,y^+]\) be two IVFNs, \(\lambda \in [0,1]\), then their operational laws are defined as follows:
\((1)~\chi +y=[\chi ^-,\chi ^+]+[y^-,y^+]=[\chi ^-+y^-,\chi ^++y^+]\);
\((2)~\lambda \chi =[\lambda \chi ^-,\lambda \chi ^+]\);
\((3)~\chi ^{\lambda } =[(\chi ^-)^{\lambda },(\chi ^+)^{\lambda }];\)
\((4)~\chi \cdot y=[\chi ^-,\chi ^+]\cdot [y^-,y^+]=[\chi ^-\cdot y^-,\chi ^+\cdot y^+]\);
\((5)~\chi =y\) iff \(\chi ^-=y^-\) and \(\chi ^+=y^+\);
\((6)~\chi ^c=[1-\chi ^+,1-\chi ^-]\);
\((7)~-\chi =[-\chi ^+,-\chi ^-]\).
Definition 3
(Son 2007) Let U be an initial universe and A be a set of parameters, a pair (F, A) is called an interval-valued fuzzy soft set over \({\widetilde{P}}(U)\) , where F is a mapping given by \(F:A\rightarrow {\widetilde{P}}(U)\). \({\widetilde{P}}(U)\) denotes the set of all interval-valued fuzzy subsets of U.
For \(\forall \varepsilon \in A, F(\varepsilon )\) is an interval-valued fuzzy subset of U, and it is called an interval-valued fuzzy value set of parameter \(\varepsilon \). Let \(F(\varepsilon )(x)\) denote the membership value that object x holds parameter \(\varepsilon \), then \(F(\varepsilon )\) can be written as an interval-valued fuzzy set that \(F(\varepsilon )=\{x/F(\varepsilon )(x)\mid x\in U\}=\{x/[F^-(\varepsilon )(x),F^+(\varepsilon )(x)]\mid x\in U\}\).
Definition 4
(Son 2007) The complement of an interval-valued fuzzy soft set (F, A), is denoted by \( (F,A)^c\) and is defined by \((F,A)^c=(F^c,A).\) \(\forall \varepsilon \in A, x\in U, F^c(\varepsilon )(x)=[1-F^+(\varepsilon )(x),1-F^-(\varepsilon )(x)]\).
For comparing the two IVFNs \((m=n=1)\), Peng and Yang (2017) derived the following comparative law-based score function.
Definition 5
(Peng and Yang 2017) For two interval-value fuzzy numbers \(\chi \) and y,
-
(1)
if \(s(\chi )>s(y)\), then \(\chi \succ y\);
-
(2)
if \(s(\chi )<s(y)\), then \(\chi \prec y\);
-
(3)
if \(s(\chi )=s(y)\), then \(\chi \sim y\).
3 A novel interval-valued fuzzy score function
This section describes some existing score functions (Peng and Yang 2017; Peng and Garg 2018) and develops a new score function by taking the discrimination power into consideration.
3.1 Some existing score functions
Definition 6
(Peng and Yang 2017) Let \(F(\varepsilon )(x)\) be an IVFN, then the score function of \(F(\varepsilon )(x)\) is defined as follows:
Definition 7
(Peng and Garg 2018) Let \(F(\varepsilon )(x)\) be an IVFN, then the score function of \(F(\varepsilon )(x)\) is defined as follows:
3.2 The novel score function
Definition 8
For an IVFN \(F(\varepsilon )(x)=[F^-(\varepsilon )(x),F^+(\varepsilon )(x)]\), the novel score function can be denoted as follows:
, where \(s_{pxd}(F(\varepsilon )(x))\in [0,1].\)
Theorem 1
For an IVFN \(F(\varepsilon )(x)=[F^-(\varepsilon )(x),F^+(\varepsilon )(x)]\), \(s_{pxd}(F(\varepsilon )(x))\) monotonically increases along with the increase of \(F^-(\varepsilon )(x)\) and \(F^+(\varepsilon )(x)\).
Proof
Based on Eq.(4), we obtain the first partial derivative of \(s_{pxd}(F(\varepsilon )(x))\) with \(F^-(\varepsilon )(x)\),
\(\frac{\partial s_{pxd}(F(\varepsilon )(x))}{\partial F^-(\varepsilon )(x)}= \frac{F^+(\varepsilon )(x)-F^-(\varepsilon )(x)}{2(1+F^+(\varepsilon )(x)-F^-(\varepsilon )(x))}\ge 0\).
Analogously, we can have the first partial derivative of \(s_{pxd}(F(\varepsilon )(x))\) with \(F^+(\varepsilon )(x)\),
\(\frac{\partial s_{pxd}(F(\varepsilon )(x))}{\partial F^+(\varepsilon )(x)}= \frac{2+F^+(\varepsilon )(x)-F^-(\varepsilon )(x)}{2(1+F^+(\varepsilon )(x)-F^-(\varepsilon )(x))}\ge 0\).
Consequently, we can obtain that \(s_{pxd}(F(\varepsilon )(x))\) monotonically increases along with the increase of \(F^-(\varepsilon )(x)\) and \(F^+(\varepsilon )(x)\). \(\square \)
Theorem 2
For an IVFN \(F(\varepsilon )(x)=[F^-(\varepsilon )(x),F^+(\varepsilon )(x)]\), the novel score function \(s_{pxd}(F(\varepsilon )(x))\) complies with the following properties.
\((1)~0\le s_{pxd}(F(\varepsilon )(x))\le 1\);
\((2)~s_{pxd}(F(\varepsilon )(x))=1\) iff \(F(\varepsilon )(x)=[1,1]\);
\((3)~s_{pxd}(F(\varepsilon )(x))=-1\) iff \(F(\varepsilon )(x)=[0,0]\).
Proof
Based on Theorem 1, we can see that if we only just take \(F^-(\varepsilon )(x)\) or \(F^+(\varepsilon )(x)\) into consideration, \(s_{pxd}(F(\varepsilon )(x))\) can have the min-value or max-value when \(F(\varepsilon )(x)=[0,0]\) or \(F(\varepsilon )(x)=[1,1]\). In other words, \(s_{pxd}(F(\varepsilon )(x))_{\min }=0\) and \(s_{pxd}(F(\varepsilon )(x))_{\max }=1\). Hence, \(0\le s_{pxd}(F(\varepsilon )(x))\le 1\). \(\square \)
Theorem 3
Let \(F(\varepsilon )(x)=(F^-(\varepsilon )(x),F^+(\varepsilon )(x))\) and \(G(\varepsilon )(x)=(G^-(\varepsilon )(x),G^+(\varepsilon )(x))\) be two IVFNs. If \(F^-(\varepsilon )(x)>G^-(\varepsilon )(x)\) and \(F^+(\varepsilon )(x)>G^+(\varepsilon )(x)\), then \(s_{pxd}(F(\varepsilon )(x))> s_{pxd}(G(\varepsilon )(x))\).
Proof
According to Theorem 1, we can find that \(s_{pxd}(F(\varepsilon )(x))\) monotonically increases along with the increase of \(F^-(\varepsilon )(x)\) and \(F^+(\varepsilon )(x)\).
Hence, if \(F^-(\varepsilon )(x)>G^-(\varepsilon )(x)\) and \(F^+(\varepsilon )(x)>G^+(\varepsilon )(x)\), then \(s_{pxd}(F(\varepsilon )(x))>s_{pxd}(G(\varepsilon )(x))\).
To test the feasibility of the proposed score function \(s_{pxd}\) for ranking IVFNs, Table 1 shows the comparison between the results achieved by the proposed score function \(s_{pxd}\) and the cases of the score functions \(s_{p1}\) and \(s_{p2}\) obtained by Peng and Yang (2017) and Peng and Garg (2018), respectively. \(\square \)
From Table 1, we can conclude that the presented score function \(s_{pxd}\) can effectively solve the deficiencies of \(s_{p1}\) proposed by Peng and Yang (2017). In other words, the presented score function can distinguish the difference of alternatives when the existing score function \(s_{p1}\) cannot deal with. Moreover, the presented score function can keep the same result as \(s_{p1}\) proposed by Peng and Garg (2018).
4 Novel operations on interval-valued fuzzy soft matrix
Matrix plays an all-important role in the broad area of science, medical diagnosis and engineering. However, the classical matrix theory cannot deal the cases involving diverse types of uncertainties. Inspired by Rajarajeswari and Dhanalakshmi (2014), we continue to mine some novel interesting operators and new properties for interval-valued fuzzy soft matrix.
Definition 9
(Rajarajeswari and Dhanalakshmi 2014) Let \(U=\{x_1,x_2,\cdots ,x_m\}\) be a universal set and E be the set of parameters given by \(E=\{\varepsilon _1,\varepsilon _2,\cdots ,\varepsilon _n\}\). Suppose that \(A\subseteq E\) and (F, A) be an IVFSS over U. Then the IVFSS can be expressed in interval-valued fuzzy soft matrix (IVFSM) form as \(I_{n\times m}=(F(\varepsilon _j)(x_i))_{n\times m}\), where
Definition 10
(Rajarajeswari and Dhanalakshmi 2014) If \(I_{n\times m}^{A}\) and \(I_{n\times m}^{B}\) be two IVFSMs, then
-
(1)
\(I_{n\times m}^{A}~\) & \(~I_{n\times m}^{B} = \left( \left[ \frac{F_A^-(\varepsilon _j)(x_i)+F_B^-(\varepsilon _j)(x_i)}{2},\frac{F_A^{+}(\varepsilon _j)(x_i)+F_B^{+}(\varepsilon _j)(x_i) }{2}\right] \right) _{n\times m}\);
-
(2)
\(I_{n\times m}^{A}~ \$ ~I_{n\times m}^{B} = \left( \left[ \sqrt{F_A^-(\varepsilon _j)(x_i)F_B^-(\varepsilon _j)(x_i)},\right. \right. \left. \left. \sqrt{F_A^{+}(\varepsilon _j)(x_i)F_B^{+}(\varepsilon _j)(x_i)}\right] \right) _{n\times m}\);
-
(3)
\(I_{n\times m}^{A}~ @ ~I_{n\times m}^{B} = \left( \left[ 2\frac{F_A^-(\varepsilon _j)(x_i)F_B^-(\varepsilon _j)(x_i)}{F_A^-(\varepsilon _j)(x_i)+F_B^-(\varepsilon _j)(x_i)},\right. \right. \left. \left. 2\frac{F_A^+(\varepsilon _j)(x_i)F_B^+(\varepsilon _j)(x_i)}{F_A^+(\varepsilon _j)(x_i)+F_B^+(\varepsilon _j)(x_i)} \right] \right) _{n\times m}\);
-
(4)
\(I_{n\times m}^{A}~ \cup ~I_{n\times m}^{B}= \left( \left[ \text {max}\{F_A^-(\varepsilon _j)(x_i),F_B^-(\varepsilon _j)(x_i)\},\right. \right. \left. \left. \text {max}\{F_A^{+}(\varepsilon _j)(x_i), F_B^{+}(\varepsilon _j)(x_i) \}\right] \right) _{n\times m}\);
-
(5)
\(I_{n\times m}^{A}~ \cap ~I_{n\times m}^{B}= \left( \left[ \text {min}\{F_A^-(\varepsilon _j)(x_i),F_B^-(\varepsilon _j)(x_i)\},\right. \right. \left. \left. \text {min}\{F_A^{+}(\varepsilon _j)(x_i), F_B^{+}(\varepsilon _j)(x_i) \}\right] \right) _{n\times m}\).
Definition 11
If \(I_{n\times m}^{A}\) and \(I_{n\times m}^{B}\) be two IVFSMs, then
-
(1)
\(I_{n\times m}^{A}~ \oplus ~I_{n\times m}^{B}= ([ F_A^-(\varepsilon _j)(x_i)+F_B^-(\varepsilon _j)(x_i)-F_A^-(\varepsilon _j)(x_i)F_B^-(\varepsilon _j)(x_i), F_A^+(\varepsilon _j)(x_i)+F_B^+(\varepsilon _j)(x_i) -F_A^+(\varepsilon _j)(x_i)F_B^+(\varepsilon _j)(x_i) ])_{n\times m}\);
-
(2)
\(I_{n\times m}^{A}~ \otimes ~I_{n\times m}^{B}= ([ F_A^-(\varepsilon _j)(x_i)F_B^-(\varepsilon _j)(x_i), F_A^+(\varepsilon _j)(x_i)F_B^+(\varepsilon _j)(x_i)])_{n\times m}\);
-
(3)
\(I_{n\times m}^{A}~ \maltese ~I_{n\times m}^{B} = \bigg (\bigg [ \frac{F_A^-(\varepsilon _j)(x_i)+F_B^-(\varepsilon _j)(x_i)}{2(F_A^-(\varepsilon _j)(x_i)+F_B^-(\varepsilon _j)(x_i)+1)}, \frac{F_A^+(\varepsilon _j)(x_i)+F_B^+(\varepsilon _j)(x_i)}{2(F_A^+(\varepsilon _j)(x_i)+F_B^+(\varepsilon _j)(x_i)+1)} \bigg ]\bigg )_{n\times m}\).
Example 1
Let \(I_{3\times 3}^{A}=\left( \begin{array}{ccc} {[}0.3,0.4] &{} [0.1,0.2] &{} [0.2,0.6] \\ {[}0.4,0.5] &{} [0.3,0.4] &{} [0.3,0.5] \\ {[}0.2,0.4] &{} [0.6,0.7] &{} [0.2,0.5] \\ \end{array} \right) \) and \(I_{3\times 3}^{B}=\left( \begin{array}{ccc} {[}0.4,0.6] &{} [0.2,0.2] &{} [0.2,0.3] \\ {[}0.2,0.3] &{} [0.3,0.5] &{} [0.4,0.6] \\ {[}0.3,0.5] &{} [0.3,0.4] &{} [0.4,0.6] \\ \end{array} \right) \) be two IVFSMs, then the operations on IVFSMs are listed in Table 2.
Lemma 1
(Peng 2019) For any number \(a,b,c\in [0,1],\) then
Theorem 4
For every interval-valued fuzzy soft matrix \(I_{n\times m}^A,\)\( I_{n\times m}^B\) and \(I_{n\times m}^C\), then
-
(1)
\((I_{n\times m}^A\oplus I_{n\times m}^B)~\) & \(~I_{n\times m}^C \subseteq (I_{n\times m}^A ~\) & \(~ I_{n\times m}^C) \oplus (I_{n\times m}^B ~\) & \(~ I_{n\times m}^C)\);
-
(2)
\((I_{n\times m}^A\otimes I_{n\times m}^B)~\) & \(~I_{n\times m}^C \supseteq (I_{n\times m}^A ~\) & \(~ I_{n\times m}^C) \otimes (I_{n\times m}^B ~\) & \(~ I_{n\times m}^C)\);
-
(3)
\((I_{n\times m}^A\oplus I_{n\times m}^B)~ \$ ~I_{n\times m}^C \subseteq (I_{n\times m}^A ~\) & \(~ I_{n\times m}^C) \oplus (I_{n\times m}^B \$ I_{n\times m}^C)\);
-
(4)
\((I_{n\times m}^A\otimes I_{n\times m}^B)~ \$ ~I_{n\times m}^C \supseteq (I_{n\times m}^A ~\) & \(~ I_{n\times m}^C) \otimes (I_{n\times m}^B \$ I_{n\times m}^C)\);
-
(5)
\((I_{n\times m}^A\oplus I_{n\times m}^B) \maltese I_{n\times m}^C \subseteq (I_{n\times m}^A \maltese I_{n\times m}^C) \oplus (I_{n\times m}^B \maltese I_{n\times m}^C)\);
-
(6)
\((I_{n\times m}^A\otimes I_{n\times m}^B) \maltese I_{n\times m}^C \supseteq (I_{n\times m}^A \maltese I_{n\times m}^C) \otimes (I_{n\times m}^B ~\maltese ~ I_{n\times m}^C)\).
Proof
We only prove the (1), (3) and (5) in detail; the (2), (4) and (6) can be proved in similar way.
(1) Let and be three given IVFSM, then
Let \(f=\frac{F_A^-(\varepsilon _j)(x_i)+F_B^-(\varepsilon _j)(x_i)+2F_C^-(\varepsilon _j)(x_i) -(F_A^-(\varepsilon _j)(x_i)+F_C^-(\varepsilon _j)(x_i))(F_B^-(\varepsilon _j)(x_i)+F_C^-(\varepsilon _j)(x_i))}{4}\) \(- \frac{F_A^-(\varepsilon _j)(x_i)+F_B^-(\varepsilon _j)(x_i)+F_C^-(\varepsilon _j)(x_i)-F_A^-(\varepsilon _j)(x_i)F_B^-(\varepsilon _j)(x_i)}{2}\), then
\(f=\frac{F_A^-(\varepsilon _j)(x_i)F_B^-(\varepsilon _j)(x_i)+F_C^-(\varepsilon _j)(x_i)\big (2 -F_A^-(\varepsilon _j)(x_i)-F_B^-(\varepsilon _j)(x_i)-F_C^-(\varepsilon _j)(x_i)\big )}{4}\ge 0\) (Lemma 1).
Hence, we can have
Similarly,
According to the proposed score function \(s_{pxd}\), we can have
(3) Let \(I_{n\times m}^A, I_{n\times m}^B\) and \(I_{n\times m}^C\) be three given IVFSM, then
Let
then
Hence, we can have
Similarly,
According to the proposed score function \(s_{pxd}\), we can have
(5) Let \(I_{n\times m}^A, I_{n\times m}^B\) and \(I_{n\times m}^C\) be three given IVFSM, then
Let
then
Hence, we can have
Similarly, we can have
According to the proposed score function \(s_{pxd}\), we can have
\(\square \)
Theorem 5
For every interval-valued fuzzy soft matrix \(I_{n\times m}^A, I_{n\times m}^B\) and \(I_{n\times m}^C\), then
-
(1)
\( (I_{n\times m}^A ~ \& ~ I_{n\times m}^B) ~\oplus ~ I_{n\times m}^C=(I_{n\times m}^A ~\oplus ~ I_{n\times m}^C) ~ \& ~ (I_{n\times m}^B ~\oplus ~ I_{n\times m}^C)\);
-
(2)
\( (I_{n\times m}^A ~ \& ~ I_{n\times m}^B) ~\otimes ~ I_{n\times m}^C=(I_{n\times m}^A ~\otimes ~ I_{n\times m}^C) ~ \& ~ (I_{n\times m}^B ~\otimes ~ I_{n\times m}^C)\);
-
(3)
\((I_{n\times m}^A ~\$~ I_{n\times m}^B) ~\oplus ~ I_{n\times m}^C\subseteq (I_{n\times m}^A ~\oplus ~ I_{n\times m}^C) ~\$~ (I_{n\times m}^B ~\oplus ~ I_{n\times m}^C)\);
-
(4)
\((I_{n\times m}^A ~\$~ I_{n\times m}^B) ~\otimes ~ I_{n\times m}^C\supseteq (I_{n\times m}^A ~\otimes ~ I_{n\times m}^C) ~\$~ (I_{n\times m}^B ~\otimes ~ I_{n\times m}^C)\);
-
(5)
\((I_{n\times m}^A ~@~ I_{n\times m}^B) ~\oplus ~ I_{n\times m}^C\subseteq (I_{n\times m}^A ~\oplus ~ I_{n\times m}^C) ~@~ (I_{n\times m}^B ~\oplus ~ I_{n\times m}^C)\);
-
(6)
\((I_{n\times m}^A ~@~ I_{n\times m}^B) ~\otimes ~ I_{n\times m}^C\supseteq (I_{n\times m}^A ~\otimes ~ I_{n\times m}^C) ~@~ (I_{n\times m}^B ~\otimes ~ I_{n\times m}^C)\);
-
(7)
\((I_{n\times m}^A ~\maltese ~ I_{n\times m}^B) ~\oplus ~ I_{n\times m}^C\supseteq (I_{n\times m}^A ~\oplus ~ I_{n\times m}^C) ~\maltese ~ (I_{n\times m}^B ~\oplus ~ I_{n\times m}^C)\);
-
(8)
\((I_{n\times m}^A ~\maltese ~ I_{n\times m}^B) ~\otimes ~ I_{n\times m}^C\subseteq (I_{n\times m}^A ~\otimes ~ I_{n\times m}^C) ~\maltese ~ (I_{n\times m}^B ~\otimes ~ I_{n\times m}^C)\).
Proof
We only prove the (1), (3), (5) and (7) in detail; the (2), (4), (6) and (8) can be proved in similar way.
(1) Let \(I_{n\times m}^A, I_{n\times m}^B\) and \(I_{n\times m}^C\) be three given IVFSM, then (3) Let \(I_{n\times m}^A, I_{n\times m}^B\) and \(I_{n\times m}^C\) be three given IVFSM, then
Let
then
Hence, we can have
Similarly, we can obtain
According to the proposed score function \(s_{pxd}\), we can have
(5) Let \(I_{n\times m}^A, I_{n\times m}^B\) and \(I_{n\times m}^C\) be three given IVFSM, then
Let
then
Hence, we can have
Similarly, we can have
According to the proposed score function \(s_{pxd}\), we can have
(7)
Let
then
Hence, we can have
Similarly, we can have
According to the proposed score function \(s_{pxd}\), we can have
\(\square \)
Definition 12
Let \(I_{n\times m}\) be an IVFSM and for every \(a,b \in N\), we define
From above definition, we can easily have the following relations:
-
(1)
If \(a\le a_1,\) then \(I_{n\times m}^{(a,b)} \supseteq I_{n\times m}^{(a_1,b)}\);
-
(2)
If \(b\le b_1,\) then \(I_{n\times m}^{(a,b)} \supseteq I_{n\times m}^{(a,b_1)}\);
-
(3)
If \(a\le a_1\) and \(b\le b_1\), then \(I_{n\times m}^{(a,b)} \supseteq I_{n\times m}^{(a_1,b_1)}\);
-
(4)
If \(I_{n\times m}^A\subseteq I_{n\times m}^B\), then \((I_{n\times m}^A)^{(a,b)} \subseteq (I_{n\times m}^B)^{(a,b)}\).
Theorem 6
Let \(I_{n\times m}\) be an IVFSM and for every \(a,b \in N\), then
-
(1)
\((I_{n\times m}^{(a,1)})^{(b,1)}=I_{n\times m}^{(ab,1)}=(I_{n\times m}^{(b,1)})^{(a,1)}\);
-
(2)
\((I_{n\times m}^{(1,a)})^{(1,b)}=I_{n\times m}^{(1,ab)}=(I_{n\times m}^{(1,b)})^{(1,a)}\);
-
(3)
\((I_{n\times m}^{(a,1)})^{(1,b)}=I_{n\times m}^{(a,b)}\);
-
(4)
\((I_{n\times m}^{(1,a)})^{(b,1)}=I_{n\times m}^{(b,a^b)}\).
Proof
It is trial. \(\square \)
Theorem 7
Let \(I_{n\times m}\) be an IVFSM and for every \(a,b,a_1,b_1 \in N\), then
-
(1)
\((I_{n\times m}^{(a,b)})^{(a_1,b_1)}=I_{n\times m}^{(aa_1,b_1b^{a_1})}\);
-
(2)
\((I_{n\times m}^{(a,b)})^{(a_1,b_1)}\otimes (I_{n\times m}^{(b,a)})^{b_1,a_1}=I_{n\times m}^{(aa_1+bb_1,a_1b_1a^{b_1}b^{a_1} )}\).
Proof
We only prove the (1), and the (2) can be proved in similar way.
(1) According to the Definition 12, we can have
\(\square \)
Theorem 8
Let \(I_{n\times m}^A\) and \(I_{n\times m}^B\) be two IVFSMs and for every \(a,b \in N\), then
-
(1)
\((I_{n\times m}^A \cap I_{n\times m}^B )^{(a,b)} \oplus (I_{n\times m}^A \cup I_{n\times m}^B )^{(a,b)} =(I_{n\times m}^A)^{(a,b)} \oplus (I_{n\times m}^B)^{(a,b)}\);
-
(2)
\((I_{n\times m}^A \cap I_{n\times m}^B )^{(a,b)} \otimes (I_{n\times m}^A \cup I_{n\times m}^B )^{(a,b)} =(I_{n\times m}^A)^{(a,b)} \otimes (I_{n\times m}^B)^{(a,b)}\);
-
(3)
\((I_{n\times m}^A \cap I_{n\times m}^B )^{(a,b)}\) & \((I_{n\times m}^A \cup I_{n\times m}^B )^{(a,b)} =(I_{n\times m}^A)^{(a,b)}\) & \((I_{n\times m}^B)^{(a,b)}\);
-
(4)
\((I_{n\times m}^A \cap I_{n\times m}^B )^{(a,b)} @ (I_{n\times m}^A \cup I_{n\times m}^B )^{(a,b)} =(I_{n\times m}^A)^{(a,b)} @ (I_{n\times m}^B)^{(a,b)}\);
-
(5)
\((I_{n\times m}^A \cap I_{n\times m}^B )^{(a,b)} \$ (I_{n\times m}^A \cup I_{n\times m}^B )^{(a,b)} =(I_{n\times m}^A)^{(a,b)} \$ (I_{n\times m}^B)^{(a,b)}\);
-
(6)
\((I_{n\times m}^A \cap I_{n\times m}^B )^{(a,b)} \maltese (I_{n\times m}^A \cup I_{n\times m}^B )^{(a,b)} =(I_{n\times m}^A)^{(a,b)} \maltese (I_{n\times m}^B)^{(a,b)}\).
Proof
We only prove the (1) in detail, the (2)–(6) can be proved in similar way.
(1) Let \(I_{n\times m}^A\) and \(I_{n\times m}^B\) be two IVFSMs, and \(\text {max}\{x,y\}+\text {min}\{x,y\}=x+y, \text {max}\{x,y\}\text {min}\{x,y\}=xy\), then
Theorem 9
Let \(I_{n\times m}^A\) and \(I_{n\times m}^B\) be two IVFSMs and for every \(a,b \in N\), then
-
(1)
\( \big ((I_{n\times m}^A)^{(a,b)}\oplus (I_{n\times m}^B)^{(a,b)}\big ) ~ \& ~ \big ((I_{n\times m}^A)^{(a,b)}\otimes (I_{n\times m}^B)^{(a,b)}\big )= (I_{n\times m}^A)^{(a,b)}~ \& ~ (I_{n\times m}^B)^{(a,b)}\);
-
(2)
\(\big ((I_{n\times m}^A)^{(a,b)}\oplus (I_{n\times m}^B)^{(a,b)}\big ) ~\maltese ~ \big ((I_{n\times m}^A)^{(a,b)}\otimes (I_{n\times m}^B)^{(a,b)}\big )= (I_{n\times m}^A)^{(a,b)}~\maltese ~ (I_{n\times m}^B)^{(a,b)}\);
-
(3)
\( [\big ((I_{n\times m}^A)^{(a,b)}\oplus (I_{n\times m}^B)^{(a,b)}\big ) \cup \big ((I_{n\times m}^A)^{(a,b)}\otimes (I_{n\times m}^B)^{(a,b)}\big )] \& [\big ((I_{n\times m}^A)^{(a,b)}\oplus (I_{n\times m}^B)^{(a,b)} \big )\cap \big ((I_{n\times m}^A)^{(a,b)}\otimes (I_{n\times m}^B)^{(a,b)}\big )] =(I_{n\times m}^A)^{(a,b)}~ \& ~ (I_{n\times m}^B)^{(a,b)}\);
-
(4)
\([\big ((I_{n\times m}^A)^{(a,b)}\oplus (I_{n\times m}^B)^{(a,b)}\big ) \cup \big ((I_{n\times m}^A)^{(a,b)}\otimes (I_{n\times m}^B)^{(a,b)}\big )] \maltese [\big ((I_{n\times m}^A)^{(a,b)}\oplus (I_{n\times m}^B)^{(a,b)} \big )\cap \big ((I_{n\times m}^A)^{(a,b)}\otimes (I_{n\times m}^B)^{(a,b)}\big )] =(I_{n\times m}^A)^{(a,b)}~\maltese ~ (I_{n\times m}^B)^{(a,b)}\);
-
(5)
\( [((I_{n\times m}^A)^{(a,b)} \otimes (I_{n\times m}^B)^{(a,b)} ) \& ( (I_{n\times m}^A)^{(a,b)} \otimes (I_{n\times m}^B)^{(a,b)} )^c] \& [((I_{n\times m}^A)^{(a,b)} \oplus (I_{n\times m}^B)^{(a,b)} ) \& ( (I_{n\times m}^A)^{(a,b)} \oplus (I_{n\times m}^B)^{(a,b)})^c]= ([\frac{1}{2},\frac{1}{2}])_{n\times m}\);
-
(6)
\( [((I_{n\times m}^A)^{(a,b)} \otimes (I_{n\times m}^B)^{(a,b)} ) \maltese ( (I_{n\times m}^A)^{(a,b)} \otimes (I_{n\times m}^B)^{(a,b)} )^c] \maltese [((I_{n\times m}^A)^{(a,b)} \oplus (I_{n\times m}^B)^{(a,b)} ) \maltese ( (I_{n\times m}^A)^{(a,b)} \oplus (I_{n\times m}^B)^{(a,b)})^c]= ([\frac{1}{4},\frac{1}{4}])_{n\times m}\).
Proof
It is trial. \(\square \)
Theorem 10
Let \(I_{n\times m}^A\) and \(I_{n\times m}^B\) be two IVFSMs and for every \(a,b \in N\), then
-
(1)
\( [ (I_{n\times m}^A \cup I_{n\times m}^B)^{(a,b)} @ (I_{n\times m}^A \cap I_{n\times m}^B)^{(a,b)} ] \$ [(I_{n\times m}^A \cup I_{n\times m}^B)^{(a,b)}\) & \((I_{n\times m}^A \cap I_{n\times m}^B)^{(a,b)}]=(I_{n\times m}^A)^{(a,b)} \$ (I_{n\times m}^B)^{(a,b)} \);
-
(2)
\([(I_{n\times m}^A)^{(a,b)} @ (I_{n\times m}^B)^{(a,b)}] \$ [(I_{n\times m}^A)^{(a,b)}\) & \((I_{n\times m}^B)^{(a,b)}] = (I_{n\times m}^A)^{(a,b)} \$ (I_{n\times m}^B)^{(a,b)}\).
Proof
We only prove the (1) in detail; the (2) can be proved in similar way.
(1) Let \(I_{n\times m}^A\) and \(I_{n\times m}^B\) be two IVFSMs, and \(\text {max}\{x,y\}+\text {min}\{x,y\}=x+y, \text {max}\{x,y\}\text {min}\{x,y\}=xy\), then
5 Novel interval-valued fuzzy soft decision-making method based on CoCoSo with CRITIC
5.1 The introduction of decision-making issue
Suppose that \(U=\{x_1,x_2,\cdots ,x_m\}\) be a set of alternatives, \(E=\{\varepsilon _1,\varepsilon _2,\cdots ,\varepsilon _n\}\) be a discrete set of parameters, and \(W=\{w_1,w_2,\cdots ,w_n\}\) be weight vector with \(w_j\in [0,1], \sum \limits _{j=1}^n w_j=1\). Suppose that the assessment of alternative \(x_i\) with respect to attribute \(\varepsilon _j\) be denoted by interval-valued fuzzy number \(F(\varepsilon _j)(x_i)=[F^-(\varepsilon _j)(x_i),F^+(\varepsilon _j)(x_i)]\), which can be shown in Table 3.
5.2 Approach to determine combined weights
5.2.1 Determine objective weights: CRITIC method
Criteria can be served as an important source of information in decision-making issues. Important parameter weights can reveal the wealth of information contained in each, called “objective weights.” The CRITIC (Criteria Importance Through Inter-criteria Correlation) is a method for calculating the objective weight of a given criterion in MCDM problems (Diakoulaki et al. 1995). The objective weights obtained through the above methods combine the strength comparison of each criterion with the conflict between the criteria. The standard intensity contrast is considered to be the standard deviation, and the conflict between them is calculated by the correlation coefficient. In this section, we extend this method to the interval value fuzzy soft environment.
Suppose that \(F(\varepsilon _j)(x_i)\) denotes the interval-valued fuzzy soft performance value of ith alternative according to jth parameter (\(i=1,2,\cdots ,m\) and \(j=1,2,\cdots ,n\)), \(w_j^o\) represents the fuzzy objective weight of jth parameter, C is a series of cost parameters, and B is a series of benefit parameters. In the following, the steps of computing interval-valued fuzzy soft objective weights-based CRITIC are listed.
Step 1: Calculate score function \(\digamma =(\tau _{ij})_{m\times n}(i=1,2,\cdots ,m; j=1,2,\cdots ,n)\) of each IVFN \(F(\varepsilon _j)(x_i)=[F^-(\varepsilon _j)(x_i),F^+(\varepsilon _j)(x_i)]\) by Eq. (6).
Step 2: Switch the matrix \(\digamma \) into a standard interval-valued fuzzy soft matrix \(\digamma '=(\tau _{ij}')_{m\times n}\) by Eq. (7)
where \(\tau _j^-=\mathop {min}\limits _i{\tau _{ij}}\) and \(\tau _j^+=\mathop {max}\limits _i{\tau _{ij}}\)
Step 3: Compute the criteria standard deviations by Eq. (8).
where \({\overline{\tau }}_j=\frac{\sum \limits _{i=1}^m\tau _{ij}'}{m}\).
Step 4: Determine the correlation between criteria pairs using Eq. (9).
Step 5: Compute the quantity of information of each criterion as follows:
The larger the \(c_j\) is, the more information a certain criterion contains, so the weight of this evaluation criterion is greater than that of other criteria.
Step 6: Calculate the objective weight of each criterion as follows:
5.2.2 Determine combined weights: linear weighted comprehensive method
Suppose that the subjective weight, given by experts, is \(w=\{w_1,w_2,\) \(\cdots ,w_n\}\), where \(\sum \limits _{j=1}^n w_j=1,0\le w_j\le 1\). The objective weight, calculated by Eq.(11), is \(\omega =\{\omega _1,\omega _2,\cdots ,\omega _n\}\), where \(\sum \limits _{j=1}^n\omega _j=1,0\le \omega _j\le 1\).
Therefore, combined weight \(\varpi =\{\varpi _1,\varpi _2,\cdots ,\varpi _n\}\) can be denoted as follows:
where \(\sum \limits _{j=1}^n \varpi _j=1,0\le \varpi _j\le 1\).
The subjective weights are unified with the objective weights by using a nonlinear weighted synthesis method. According to multiplier effect, the larger the value of subjective weight and objective weight, the larger the combined weight, and vice versa. In additional, it is easily seen that Eq. (12) promotes restrictions that only consider objective or subjective effects. The advantage of Eq. (12) is that the parameter weights of alternatives can simultaneously show objective information and subjective information (Wang and Li 2020; Lin et al. 2020).
5.3 The interval-valued fuzzy soft CoCoSo method
CoCoSo (Combined Compromise Solution) is a new and resultful MCDM method, which is explored by Yazdani et al. (2019) recently. It has successfully applied in diverse domains such as hospital service quality (Roy et al. 2018), location selection (Yazdani et al. 2019, 2020), stock management (Erceg et al. 2019), sustainability assessment (Ecer et al. 2020) and financial risk evaluation (Peng and Huang 2020). The presented approach is based on an integrated exponentially weighted product (EWP) and simple additive weighting (SAW) model, which can be a compendium of compromise solutions. In order to deal the MCDM issue, we develop an IVFS-CoCoSo approach.
Generally speaking, the IVFS-CoCoSo method has the following steps.
Remark 1
It should be noted that the interval-value fuzzy soft CoCoSo method whose information in the decision matrix is denoted by IVFNs to express the preference of DMs. IVFN is very effective in capturing the uncertainties and inaccuracies of experts or DM in MCDM problems. In addition, the IVFS-CoCoSo method is a valuable tool for dealing with DM problems, which has a strong ability to distinguish optimal alternative from nominated alternatives, without counterintuitive phenomena, and no division by zero problem (Peng and Garg 2018). Nevertheless, others MCDM methods based on interval-valued fuzzy soft environments have no such valuable features. The framework of proposed method is presented in Fig. 1.
The framework for using the proposed method
6 A case study in intelligent health management evaluation
Intelligent health management (IHM) is a new industry and new format formed by the integration of health management, artificial intelligence, big data and other new generation of information technology (Fig. 2).
The framework of intelligent health management
6.1 The development status of health management
Most people’s understanding of health management (Mardani et al. 2019) has some misunderstanding as follows:
-
(1)
Health management is considered to be a higher-level healthcare program for the economically well-off, which is an additional burden for the economically less well-off.
However, the fact is: the experience of more than 20 years’ study on health management in the USA shows that there is a relationship between 90% and 10% of health management for any enterprise and individual. China is no exception, with the least healthy 1% of the population and the 19% with chronic diseases sharing 70% of medical expenses. Therefore, accurate understanding of their health status and potential risks through health management can not only effectively reduce the risk of disease, but also save medical expenses.
-
(2)
It is believed that health management is only for the provision of services to the sick groups in medical institutions who need special care.
However, in fact, compared with interventional therapy, health management pays more attention to “preventing diseases before they happen,” and determines the different emphasis of health management according to the characteristics of different populations and different health risks. Therefore, the main service objects of health management are the healthy and sub-healthy people in the society, as well as the groups with chronic diseases and diseases before recovery.
-
(3)
Health management is equal to physical examination, equal to the healthcare products seen everywhere in the market, that there is no necessary connection with disease prevention.
But in fact, the WHO study data show that one-third human disease through preventive health care can be avoided, one-third can be effectively controlled through early discovery, one-third by information communicate effectively improve therapeutic effect. Therefore, the occurrence of all kinds of diseases has its certain reasons and rules, which are determined by congenital genetic factors and influenced by acquired behaviors and lifestyles.
Therefore, this paper believes that the meaning of health management contains three characteristics: (1) it is a process of overall management of individual or group health risk factors; (2) it is a non-medical means to help people approach the level of complete physical and mental health; (3) it can mobilize individual and collective initiative, effectively use limited resources to achieve the maximum health improvement effect.
Countries around the world have introduced laws and policies related to health management industry and held relevant competitions to promote the rapid development of the industry. Health management services in China are still dominated by physical examination, and the incidence of chronic diseases remains high (Fig. 3), showing the characteristics of large number of patients and high medical costs. According to the blue book of health management 2018, the number of outpatients in China’s medical services was estimated to exceed 8 billion in 2017, while the number of health management (physical examination) patients was only 500 million. Moreover, the health management service is single. More than 95% of the health management service is still physical examination, lacking of post-examination service. According to the statistical bulletin of health development in China in 2017, the total health expenditure reached 5159.88 billion Yuan in 2017, accounting for 6.2% of the GDP. Compared with 2010, the compound annual growth rate was as high as 15%, and the total health expenditure showed a rapid growth trend.
Types of chronic diseases in Chinese physical examination population
6.2 The development status of intelligent health management
1. Combination of artificial intelligence and health management
-
(1)
Big data and flu prediction (health management + big data)
Methods: establish related database, intelligent analysis model, etc.
-
(2)
Machine learning and blood glucose management (health management + machine learning algorithm)
Methods: accurate diabetes model was established by machine learning algorithm and other techniques.
-
(3)
Database technology and health factor detection (artificial intelligence + genotype + health management)
Methods: establish health big data platform, database, etc.
-
(4)
Improvement of health management and quality of life (artificial intelligence algorithm + big data + health management)
Methods: health management optimization platform and health promotion scheme were established.
2. Policy, market and other factors drive the development of intelligent health management industry
The accelerated aging of the population, the introduction of national policies and the improvement of artificial intelligence technology will promote the intelligent and personalized development of health management.
-
(1)
The aging of the population
According to data released by the national bureau of statistics, 249.49 million people aged 60 or above were born in 2018, 8.59 million more than the previous year, accounting for 17.9% of the total population. Among them, the population aged 65 or above accounted for 11.9% of the total population, an increase of 8.27 million over the previous year. According to the 2017 world population outlook released by the United Nations, the number of people aged 60 and above will double by 2050 and more than triple by 2100, with 65% of them in Asia.
-
(2)
Health management services have great potential in the future
While 70% of people in the USA have access to health management services, less than 0.1% of people in China have access to health management services. As China’s economy continues to improve, the demand for health management services will expand.
-
(3)
State policy support and industry planning
In 2017, the state council issued the notice on the development plan of a new generation of artificial intelligence, which specifically pointed out to strengthen group intelligent health management, break through key technologies such as health big data analysis and Internet of things, and promote the transformation of health management from point-like monitoring to continuous monitoring, and from short process management to long process management.
-
(4)
The promotion of artificial intelligence technology
The deep learning algorithm based on deep convolution neural network accelerates the development of artificial intelligence technology and promotes the cross-border integration between health management and it.
Health management services include health record management, lifestyle management, dynamic tracking management and many other services (Fig. 2). Currently, the main services in the market are only health checkup management and disease management.
In the process of evaluating the health checkup of people, it is very necessary to design a reasonable assessment system to guarantee the effective and scientific evaluation results (Shi et al. 2018). This section constructs and depicts an evaluation criteria of health checkup as \(\varepsilon _j (j = 1, 2, \cdots , 16)\). The description of each parameter is briefly stated in Table 4.
Parameter comprehensive evaluation model is realized by establishing parameter scoring system. Whether the index score is accurate and reasonable, the key is to find sensitive and specific grade cut points. Here, the epidemiological data of the above 16 main parameters are sorted and summarized, and the latest general normal value standards of clinical internal medicines are reviewed. Various types of cardiovascular disease, hypertension, dyslipidemia, diabetes, obesity prevention and treatment guidelines, and the cutoff points provided in various clinical diagnostic criteria, the cut-point suitability is measured by the percentage of attributable risk of different cut-point populations. Find the point where the sensitivity and specificity are relatively good and the false positive rate is relatively low. The principle of the health management process is shown in Fig. 4.
The principle of health management process cut-point on Health-Sub-health-Onset
According to the health management procedure principle of Fig. 4, the selected 16 important physiological parameters are classified into five levels according to health (excellent), sub-health (general), alert (raise), symptoms (early disease) and onset (disease). The parameter score value is interval-valued fuzzy set to [0,1], and the 5 levels are assigned [0,0.2], [0.2,0.4], [0.4,0.6], [0.6,0.8] and [0.8,1] by the equalization assignment method.
First, the latest general-purpose parameter normal value is used as the benchmark value of each parameter, and it is established as an excellent health [0.8,1] cut point. Then, the key research parameters raised the alert and the disease occurred at two cut points. The warning of the increase in the index is sub-health downstream. The significance of the cutoff point is to provide a boundary that should be vigilant and begin to intervene, prevent the risk factors of most chronic diseases from rising early and do not cause excessive psychological pressure on health check individuals. The margins of the parameters proposed in various guidelines are elevated, and the risk of related diseases is slightly increased, but the clinical diagnostic criteria are not met. This category belongs to this level, which is in the reversible stage of the body’s physiological compensation, such as the normal high value of hypertension, blood sugar, damage and elevated blood lipids. The division of the disease occurrence point is based on the principle that the measured value of the parameter can clarify the diagnosis of the relevant disease, and the disease process has passed the pre-disease stage, and the body may have a qualitative pathological damage, such as hypertension level 2, moderate anemia, obesity, symptoms, hyperlipidemia, severe liver and kidney function damage. The parameter is between the rising point of warning and the point at which the disease occurs. It is generally in the preclinical stage of the disease. Clinical symptoms may occur, and the risk of related diseases is moderately increased. It is classified as the early cut point of the disease. According to the above principles, the scores of important physiological parameters for health check-up are presented. The detailed data are shown in Table 5.
Example 2
Suppose that there are six individuals \(U =\{x_1,x_2, x_3, x_4, x_5, x_6\}\) to be considered for health check-up. The parameter set \(E= \{ \varepsilon _1,\varepsilon _2,\varepsilon _3,\varepsilon _4,\varepsilon _5,\varepsilon _6,\varepsilon _7,\varepsilon _8,\) \(\varepsilon _9,\varepsilon _{10},\varepsilon _{11},\varepsilon _{12},\varepsilon _{13},\varepsilon _{14},\varepsilon _{15},\varepsilon _{16} \}\) is employed in assessing the individuals by advanced AI equipments, which \(\varepsilon _1\) (body mass index), \(\varepsilon _2\) (waist circumference), \(\varepsilon _3\) (heart rate), \(\varepsilon _4\) (blood pressure), \(\varepsilon _5\) (electrocardiogram), \(\varepsilon _6\) (pulmonary X-ray), \(\varepsilon _7\) (white blood cell count), \(\varepsilon _8\) (hemoglobin), \(\varepsilon _9\) (transaminase), \(\varepsilon _{10}\) (total bilirubin), \(\varepsilon _{11}\) (blood creatinine), \(\varepsilon _{12}\) (urinary protein), \(\varepsilon _{13}\) (lipids), \(\varepsilon _{14}\) (blood glucose), \(\varepsilon _{15}\) (homocysteine) and \(\varepsilon _{16}\) (uric acid). There is no need to distinguish the so-called benefit parameters and cost parameters because all parameters are converted to corresponding IVFNs with uniform scale. Based on years of health check-up experience and mature technology, the weight information is assigned as \(W=(0.04, 0.05, 0.03, 0.06, 0.1, 0.05, 0.07, 0.1, 0.04, 0.08, 0.12, 0.06, 0.04, 0.06, 0.05, 0.05)\).
The assessments for health check-up individuals arising from the medical machine detection employing advanced technology and the medical examination data with its tabular form presented in Table 6.
Next, we employ the developed algorithm (\(\lambda =0.5\)) to choose the healthiest check-up individual under interval-valued fuzzy soft information.
Step 1. Obtain the interval-valued fuzzy soft set (F, A) employing the 5 levels by Table 5, as shown in Table 7.
Step 2. Compute the score function \(\digamma =(\tau _{ij})_{6\times 16} \) of each IVFN \(F(\varepsilon _j)(x_i)=[F^-(\varepsilon _j)(x_i),F^+(\varepsilon _j)(x_i)]\) by Eq. (6).
Step 3. There is no need to switch the matrix due to all parameters are benefit parameters.
Step 4. Calculate combined weight \(\varpi \) by Eq. (12) as follows:
\(\varpi _1=0.0367,\varpi _2=0.0436,\varpi _3=0.0354,\varpi _4=0.0626,\)
\(\varpi _5=0.0918,\varpi _6=0.0459,\varpi _7=0.0669,\varpi _8=0.0918,\)
\(\varpi _9=0.0349,\varpi _{10}=0.0852,\varpi _{11}=0.1278,\varpi _{12}=0.0626,\)
\(\varpi _{13}=0.0382,\varpi _{14}=0.0573,\varpi _{15}=0.0522,\varpi _{16}=0.0671\).
Step 5. Compute the total of the weighted comparability sequence for every check-up individual as \(S_i\):
\(S_1=0.9823,S_2=0.8226,S_3=0.6096,S_4=0.4472,S_5=0.2661,S_6=0.0671.\)
Step 6. Compute the whole of the power weight of comparability sequences for each check-up individual as \(P_i\):
\(P_1=15.9758,P_2=13,P_3=11,P_4=8,P_5=4,P_6=1.\)
Step 7. Three appraisal score strategies are used to generate relative weights of other options, which are derived using Eqs. (15)-(17) and shown as follows:
\(k_{1a}=0.3019, k_{2a}=0.2461, k_{3a}=0.2067, k_{4a}=0.1504, k_{5a}=0.0759, k_{6a}=0.019.\)
\(k_{1b}=30.6106, k_{2b}=25.2562, k_{3b}=20.0827, k_{4b}=14.6628, k_{5b}=7.9651, k_{6b}=2.\)
\(k_{1c}=1.0000, k_{2c}=0.8151, k_{3c}=0.6846, k_{4c}=0.4981, k_{5c}=0.2516, k_{6c}=0.0629.\)
Step 8. Compute the assessment value \(k_i\) by Eq. (18) as follows:
\(k_1=12.736,k_2=10.4899,k_3=8.4077,k_4=6.1355,k_5=3.2981,k_6=0.8277.\)
Step 9. Rank the check-up individuals according to the decreasing values of assessment value \(k_i\) as follows:
\(x_1\succ x_2\succ x_3\succ x_4\succ x_5\succ x_6\).
6.3 The analysis of weight information
In the following, we give some comparisons of the original weight, the objective weights (Yang and Peng 2017; Xiao et al. 2013; Chen and Zou 2017), the existing combined weights (Peng et al. 2017; Peng and Garg 2018) and the proposed combined weights.
From Fig. 5, the developed combined weights determining method can both availably reveal the objective preference (CRITIC) and subjective preference to some extent. For objective weights proposed by Yang and Peng (2017), the objective weights \(w_3,w_5,w_8,w_9,w_{11}\) and \(w_{13}\) are almost half or double of the original weight values, which may lead to diverse ranking results or optimal alternative in the process of decision making. The objective weights, proposed by Xiao et al. (2013), encounter the same situation for \(w_3,w_4\) (double or more) and \(w_5,w_7,w_8,w_{11}\) (half or less). Moreover, the objective weights in Xiao et al. (2013) are not actually true due to the number m of alternatives is less than the number n of parameters. In other words, the equation \(c_{ij}=\sum _{k=1}^n (f_{ik}-f_{jk})(i=1,2,\cdots ,m;j=1,2,\cdots ,n)\) in Definition 8 (Xiao et al. 2013) is no way to calculate. For the objective weights, proposed by Chen and Zou (2017), encounter the same situation for \(w_3\) (double or more) and \(w_5,w_8,w_{11}\) (half or less). As a consequence, the proposed combined weights not only have a certain difference compared with original weights, but also have kept it within reasonable limits.
The comparison of the weight information
The comparison of the weight information (Peng and Garg 2018)
From Fig. 6, the combined weights, proposed by Peng and Garg (2018), are gradually keep a relatively steady state (tend to some fixed values) when p increases. For combined weight \(w_{11}\), the weight values are almost 4 times different at first. Although it gradually tends to 2 times different, it also may lead to diverse ranking results or optimal alternative in the process of decision making. Hence, the existing combined weights (Peng and Garg 2018) fail to obey the rule of differentiation within a certain range. However, the proposed combined weights have no such issue. Moreover, it is also hard to choose a suitable parameter value p for obtaining optimal alternative and ranking.
The comparison of the weight information (Peng et al. 2017)
From Fig. 7, the combined weights, proposed by Peng et al. (2017), are almost the same as the original weights with very subtle changes. Especially, the combined weights are almost approaching to the original weights when q is increasing. In other words, it cannot reflect the idea of objective weights. But for our proposed combined weights, they can have appropriate difference, which take the objective information into consideration compared with the original weights.
7 Comparison with some existing methods
7.1 The discrimination degrees of some existing methods
For a better comparison with some existing methods (Son 2007; Yang and Peng 2017; Yang et al. 2009; Yuan and Hu 2012; Xiao et al. 2013; Peng et al. 2017; Chen and Zou 2017; Peng and Garg 2018), the comparison results are shown in Fig. 8 by employing the Example 2.
For some existing decision-making methods [TOPSIS (Yang and Peng 2017), Choice value-1 (Yang et al. 2009), CODAS (Peng et al. 2017)], they have stronger power of discrimination degrees during the process of decision making. However, the other parts of decision-making methods [WDBA (Peng and Garg 2018), similarity measure-1 (Peng and Garg 2018), MABAC (Peng et al. 2017), EDAS (Peng et al. 2017), similarity measure-2 (Peng et al. 2017), choice value-2 (Son 2007), GRA (Chen and Zou 2017)] fail to such advantages, which make the DMs or experts hardly choose the optimal health check-up individual in convincing and resultful way. Moreover, it is easily known that the proposed MCDM method-based CoCoSo possesses high degree of differentiation compared with some existing MCDM methods.
The comparison of the existing decision-making methods
7.2 The division by zero problem of some existing methods
Example 3
Suppose that there have another medical machine detection employing advanced technology and the medical examination data with its tabular form presented in Table 8.
Remark 2
From Table 9, we can find the conclusive ranking and optimal health check-up individual by the developed method are in agreement with the decision results of WDBA (Peng and Garg 2018), CODAS (Peng and Garg 2018), similarity measure (Peng and Garg 2018), MABAC (Peng et al. 2017), EDAS (Peng et al. 2017), similarity measure (Peng et al. 2017), choice value-1 (Yang et al. 2009), comparison table (Son 2007; Yuan and Hu 2012), and GRA (Chen and Zou 2017). In additional, for the TOPSIS (Yang and Peng 2017), they fail to achieve the optimal health check-up individual and ranking order because the “division by zero problem.”
7.3 The counterintuitive phenomena of some existing methods
Example 4
Suppose that there have another medical machine detection employing advanced technology and the medical examination data (physical examination items (\(\varepsilon _3,\varepsilon _4\) and \(\varepsilon _7\) with weight \(w=(0.3,0.3,0.4)\)) specified by health check-up individuals) with its tabular form presented in Table 10.
Remark 3
From Table 11, it can be easily seen that the optimal health check-up individual computed by the developed method is in agreement with the decision results of WDBA (Peng and Garg 2018), CODAS (Peng and Garg 2018), similarity measure-1 (Peng and Garg 2018), MABAC (Peng et al. 2017), EDAS (Peng et al. 2017), similarity measure-2 (Peng et al. 2017). For TOPSIS (Yang and Peng 2017) and GRA (Chen and Zou 2017), the optimal health check-up individuals are all \(x_1\), which is different from the proposed method. The main reason is that the weight determining method influence the final results. Moreover, we also find that the Choice value-1 (Yang et al. 2009) and Comparison table (Son 2007; Yuan and Hu 2012) cannot determine the optimal health check-up individual due to their drawbacks of counterintuitive phenomena, which has been discussed in Peng and Garg (2018).
8 Conclusion
The key contributions can be concluded below.
-
(1)
The new score function for IVFN is proposed, which has strong power in distinguishing two IVFNs;
-
(2)
The combined weight model is proposed based on CRITIC and linear weighted comprehensive method, which can simultaneously consider subjective preference and objective preference;
-
(3)
Some matrix operations on IVFSSs are presented and their interesting properties are proved in detail;
-
(4)
The novel interval-valued fuzzy soft MCDM method based on CoCoSo is proposed, which can obtain the best alternative without counterintuitive phenomena, achieve the decision results without division by zero problem and possess a strong ability to differentiate the optimal alternative.
However, the proposed MCDM method consumes a certain amount of time complexity, and lacks the self-adjusting weight mechanism (Huang and Liang 2019). How to reduce the time complexity and fuse the idea of self-paced learning are key research directions in the future. Moreover, the outstanding CoCoSo method for handling the intelligent healthcare management decision-making issues under diverse fuzzy environment (Shen et al. 2020; Zhan and Alcantud 2019; Alcantud et al. 2019; Wang et al. 2020) is an interesting topic.
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Acknowledgements
The authors are very appreciative to the reviewers for their precious comments which enormously ameliorated the quality of this paper. Our work is sponsored by the National Natural Science Foundation of China (No 61806213), MOE (Ministry of Education in China) Project of Humanities and Social Sciences (No. 18YJCZH054), Natural Science Foundation of Guangdong Province (Nos. 2018A030307033, 2018A0303130274), Social Science Foundation of Guangdong Province (No. GD18CFX06) and Special Innovation Projects of Universities in Guangdong Province (No. KTSCX205).
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Peng, X., Krishankumar, R. & Ravichandran, K.S. A novel interval-valued fuzzy soft decision-making method based on CoCoSo and CRITIC for intelligent healthcare management evaluation. Soft Comput 25, 4213–4241 (2021). https://doi.org/10.1007/s00500-020-05437-y
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DOI: https://doi.org/10.1007/s00500-020-05437-y