Smart algorithmic solutions for neutrosophic multiple-attribute decision-making and applications to chair furniture comfort design evaluation

Abstract

The activities in public places are becoming increasingly frequent, and public seats have become an important component of public space and an “auxiliary tool” for work, life, and leisure. The demand for such furniture in China is gradually increasing, but the design and research work of public seats is lagging behind the development status and speed of other civil and office furniture. Usually, public seats can be divided into two parts based on their usage: public indoor seats and public outdoor seats. How to understand the role and impact of public seats on the indoor environment, and how to establish a new design method for public seats in the context of the overall environmental system, are currently urgent issues to be solved. The chair furniture comfort design evaluation is a classical MADM issues. In such paper, the generalized weighted Bonferroni mean (WBM) operator and power average (PA) is constructed for MADM with single-valued neutrosophic sets (SVNSs). Then, the generalized single-valued neutrosophic number power WBM (GSVNNPWBM) operator is built and then the MADM decision methods are proposed based on the GSVNNPWBM operator. Finally, an example about chair furniture comfort design evaluation and some comparative analysis were given to demonstrate the GSVNNPWBM method.

Keywords

Multiple attribute decision making (MADM)single-valued neutrosophic sets (SVNSs)weighted Bonferroni mean (WBM) operator power average (PA)chair furniture comfort design evaluation

1. Introduction

Public seats are an intermediary between people and the spatial environment: people furniture spatial environment [1, 2]. Public furniture design has the following major characteristics based on different spatial functions and social activity content: (1). Strong professionalism. Some specific public spaces require seating, and public seats have special requirements for each part of the seat due to the functional needs of the specific space. For example, the furniture design of aircraft, cars and ships is the furniture type with the highest high-tech content in furniture design. The design of a Airline seat has become a multidisciplinary, complex system engineering design, which is impossible for one person to complete. It needs to be created under a number of special conditions such as full cost, comfort, strict aviation safety regulations, etc. Usually, world-renowned or aspiring design firms take pride in undertaking such design businesses [1, 2, 3, 4]. For example, British Airways selected Tangerine Design Company from more than 30 design companies around the world to redesign refreshing business travel facilities after five rounds of bidding activities. The cost of this project is 200 million pounds. The designers of the design company, the internal Brand management, cabin service, design management, engineering technology, product development, supply and many other personnel of the company, as well as the external ergonomics experts, doctors, the Royal Air Force and seat manufacturers, participated in all aspects of the design, and completed the design task after nearly a year and a half. The final finished product officially put into production. The strong professionalism of public seats is evident [5, 6]. (2). There are fewer types. Due to the public and communicative characteristics of public seats, they must cater to the aesthetic views of the majority of people and adapt to a wider audience while meeting their basic functions. Thus, to some extent, there is a fixed pattern limitation on the color, texture, and connotation of its function, form, and materials. Thus, the differentiation of this type of furniture is relatively small compared to other furniture categories. Similar or similar situations often occur. There are few types, and many series products only make a few minor adjustments in details [7, 8]. (3). Large quantity. Due to its adaptability to public activities, furniture must be able to be used by a certain group of people. So it usually does not appear in a single piece or group of furniture, but in a row or groups. And it is precisely because of the large number of common seats that the space becomes flesh and blood, the order of the space is determined, and the connotation of the space is enriched [9, 10]. (4). Privacy and interactivity coexist. With the increasing competition in the industry, everyone in the industry is required to take every challenge seriously. More comfortable and luxurious means greater profit margins, while strong interactivity and good communication mean high-quality spiritual enjoyment [11, 12]. People are willing to pay for this kind of thoughtfulness, so exploring the best way to utilize space between private space and interactivity, as well as the human factor of feeling comfortable, has become a characteristic of the future development of this type of furniture. Public indoor seats provide convenience for people’s work, life, leisure and other activities in public places [13, 14]. They not only have the common characteristics of ordinary seats, but also have their own characteristics of strong professionalism, few types, large quantity, interactivity, and privacy coexistence [15, 16]. This provides us with other perspectives in product development besides considering factors such as easy production, timely delivery, and competitive costs. Actions speak louder than words. On the one hand, the design practice of public indoor seats can be creatively improved and designed from the seats themselves, in an attempt to create a new way of life, form a new feeling, and create a more appropriate sense of space [17, 18]; On the other hand, the integration and trade-off of various design elements can be considered from the perspective of the coordination between public indoor seats and the environment, as well as creating a harmonious and quiet atmosphere. The comfort design evaluation of chair furniture involves a comprehensive evaluation of multiple factors, indicators, and levels [19, 20]. The selection of evaluation indicators has characteristics such as completeness and non- overlap. Based on user experience theory, this article divides the comfort design evaluation of chair furniture into four levels: visual perception of comfort, manufacturing cost of comfort, usage perception of comfort, and emotional experience of comfort [21, 22]. These four levels have independence, non-overlap, integrity, and progressive experience levels. Chair furniture forms a texture (including visual texture, tactile texture, and emotional texture) through the regular combination of materials, form, color, and structure, which can better meet the sitting function and user behavior planning [23]. And its comfort design is transmitted to the user’s visual perception experience through external features such as the shape, color, material, and structural form of the seat [24, 25]. Through the physical properties of the material, the interface angle formed by the backrest and seat surface, and the human-machine size of the seat, the user’s perception experience is transmitted. Due to the influence of geography, culture, experience, work, and other factors on the type and role of people, there are different emotional needs for seat comfort design. The comfort of chair furniture is mainly achieved through the physiological and psychological reactions that occur during sitting, forming a comprehensive emotional experience [26, 27]. People should not only pay attention to the physiological and psychological experience of seat comfort, but also pay attention to the visual experience generated by the environment and design elements of the seat, and be able to meet the emotional needs of different user groups. In the comfort design process of chair furniture, the seat conveys its sitting comfort to users through its shape characteristics, color matching, material texture, structural form, and technological level (surface characteristics), while users obtain their comfort level through visual perception [28]. The physical characteristics of the backrest and seat surface tilt angle, as well as material comfort, are used to convey the sitting comfort, while users obtain their comfort level through their physiological perception during use [29, 30]. The comprehensive experience of the surface and physical characteristics of the seat is to meet the higher-level needs of users, namely emotional experience needs, which are both independent and interconnected [31, 32]. The comfort design evaluation indicators for chair furniture are divided into four levels: visual perception of comfort, manufacturing cost of comfort, usage perception, and emotional experience. A relatively scientific and comprehensive comfort design evaluation index system is established, and the maximum deviation method-information fusion method is used to solve the problem of subjective assignment in existing design evaluation methods, achieving objectification and quantification of the evaluation process.

The MADM is an important research area in modern management science [33, 34, 35, 36, 37]. Similar to MADM, multi-criteria decision analysis (MCDA) also plays an important role within the decisionmaking problem [38, 39, 40, 41]. There are more and more MCDA models are employed to solve the decision issues, such as, Characteristic Objects Method (COMET) method [42, 43], Data vARIability Assessment TOPSIS (DARIA-TOPSIS method) [44] and RANking COMparison (RANCOM) method [45]. In recent years, the research on MADM has attracted great attention from scholars at home and abroad [46, 47, 48, 49, 50]. Zadeh [51] constructed the fuzzy sets (FSs). Atanassov [52] constructed the intuitionistic fuzzy sets (IFSs). Smarandache [53] constructed the neutrosophic sets (NSs). Wang et al. [54] constructed the SVNSs. Chakraborty and Saha [55] constructed the selection of forklift unit for transport handling using integrated MCDM under neutrosophic environment. Broumi et al. [56] constructed the complex fermatean neutrosophic graph for decision making. Liu and Yang [57] constructed the EDAS method for physical education teaching quality evaluation. Chakraborty et al. [58] constructed the cylindrical neutrosophic single-valued number in networking problem. Haque et al. [59] constructed the novel logarithmic operational law and aggregation operators for trapezoidal neutrosophic number with MCGDM. Chakraborty et al. [60] constructed the trapezoidal bipolar neutrosophic number, de-bipolarization technique in cloud service-based MCGDM problem. Jafar et al. [61] constructed the distance and similarity measures using Max-Min operators of neutrosophic hypersoft Sets with Site Selection for Solid Waste Management Systems. The chair furniture comfort design evaluation is a classical MADM issues. However, in realistic MADM problems, due to the existence of input variables with the property of inter-influence and inter-dependence, which is not taken into account in the above studied integration operator [62, 63, 64, 65, 66]. To overcome this drawback, the Bonferroni Mean (BM) [67] operator is used as an integration operator with the property of being able to reflect the interrelationship between data. Subsequently, Yager [68] proposed the generalized BM (GBM) operator and applied it to MADM. Zhu et al. [69] proposed geometric BM (GBM) with BM and geometric mean (GM). Xia et al. [70] built the generalized WBM operator. In such paper, connected the generalized WBM operator [70] with PA operator [71] is designed for solving the MADM under SVNSs. The GSVNNPWBM operator is built and then the MADM methods are proposed based on the GSVNNPWBM operator. Eventually, an example about chair furniture comfort design evaluation and some comparative analysis were given. The main research aim and motivation of this paper is constructed: (1) The maximizing deviation method [72] is employed to determine the weight under SVNSs; (2) the generalized weighted Bonferroni mean (WBM) operator and power average (PA) is extended to SVNSs; (3) the generalized single-valued neutrosophic number power WBM (GSVNNPWBM) operator is built to solve the MADM issue; (4) Finally, an example about chair furniture comfort design evaluation and some comparative analysis were given to demonstrate the GSVNNPWBM operator.

In order to do so, the reminder of this paper proceeds. The SVNSs is concisely reviewed in Sec. 2. The GSVNNPWBM operator is built in Sec. 3. The MADM based on the GSVNNPWBM operator is built in in Sec. 4. An example application for chair furniture comfort design evaluation is given in Sec. 5. The conclusion is listed in Sec. 6.

2. Preliminaries

Wang et al. [54] structured the SVNSs.

Definition 1 [54]. The SVNSs $Z A$ in $X$ is structured:

$\displaystyle ZA=\{{({x,ZT_{A}(x),ZI_{A}(x),ZF_{A}(x)})|{x\in X}}\}$ (1)

with the truth-membership $ZT_{A}(x)$ , indeterminacy-membership $ZI_{A}(x)$ and falsity-membership $ZF_{A}(x)$ , $ZT_{A}(x),ZI_{A}(x),ZF_{A}(x)\in[{0,1}]$ , $0\leqslant ZT_{A}(x)+ZI_{A}(x)+ZF_{A}(x)\leqslant 3$ .

The SVNN is structured as $ZA=({ZT_{A},ZI_{A},ZF_{A}})$ , where $ZT_{A},ZI_{A},ZF_{A}\in[{0,1}]$ , and $0\leqslant ZT_{A}+ZI_{A}+ZF_{A}\leqslant 3$ .

Definition 2 [73]. Let $ZA=({ZT_{A},ZI_{A},ZF_{A}})$ , the score value (SV) is constructed:

$\displaystyle SV({ZA})=\frac{({2+ZT_{A}-ZI_{A}-ZF_{A}})}{3},SV({ZA})\in[{0,1}].$ (2)

Definition 3 [73]. Let $ZA=({ZT_{A},ZI_{A},ZF_{A}})$ , the accuracy value (AV) is constructed:

$\displaystyle AV({ZA})=ZT_{A}-ZF_{A},AV({ZA})\in[{-1,1}].$ (3)

Peng et al. [73] structured the order relation.

Definition 4 [73]. Let $ZA=({ZT_{A},ZI_{A},ZF_{A}})$ , $ZB=({ZT_{B},ZI_{B},ZF_{B}})$ , let $SV({ZA})=\frac{({2+ZT_{A}-ZI_{A}-ZF_{A}})}{3}$ and $SV({ZB})=\frac{({2+ZT_{B}-ZI_{B}-ZF_{B}})}{3}$ , and let $AV({ZA})=ZT_{A}-ZF_{A}$ and $AV({ZB})=ZT_{B}-ZF_{B}$ , if $SV({ZA})<SV({ZB})$ , $ZA<ZB$ ; if $SV({ZA})=SV({ZB})$ , (1) if $AV({ZA})=AV({ZB})$ , $ZA=ZB$ ; (2) if $AV({ZA})<AV({ZB})$ , $ZA<ZB$ .

Definition 5 [74]. Let $ZA=({ZT_{A},ZI_{A},ZF_{A}})$ and $ZB=({ZT_{B},ZI_{B},ZF_{B}})$ , the distance measure is defined:

$\displaystyle DM({ZA,ZB})=\frac{|{ZT_{A}-ZT_{B}+ZI_{A}-ZI_{B}}|}{4}+\frac{|{ZT% _{A}-ZT_{B}+FI_{A}-FI_{B}}|}{4}$ (4)

Definition 5 [54]. Let $ZA=({ZT_{A},ZI_{A},ZF_{A}})$ and $ZB=({ZT_{B},ZI_{B},ZF_{B}})$ , the structured operations are:

$\displaystyle(1)\;ZA\oplus ZB=({ZT_{A}+ZT_{B}-ZT_{A}ZT_{B},ZI_{A}ZI_{B},ZF_{A}% ZF_{B}});$ $\displaystyle(2)\;ZA\otimes ZB=({ZT_{A}ZT_{B},ZI_{A}+ZI_{B}-ZI_{A}ZI_{B},ZF_{A% }+ZF_{B}-ZF_{A}ZF_{B}});$ $\displaystyle(3)\;\lambda ZA=({1-({1-ZT_{A}})^{\lambda},({ZI_{A}})^{\lambda},(% {ZF_{A}})^{\lambda}}),\lambda>0;$ $\displaystyle(4)\;({ZA})^{\lambda}=({({ZT_{A}})^{\lambda},({ZI_{A}})^{\lambda}% ,1-({1-ZF_{A}})^{\lambda}}),\lambda>0.$

Definition 6 [70]. Let $p,q,r\geqslant 0$ , $xa_{i}(i=1,2,3,\ldots,n)$ be positive numbers with weight $zw=(zw_{1},zw_{2},\ldots,zw_{n})^{T}$ , $zw_{i}\in[{0,1}]$ , $\sum\nolimits_{i=1}^{n}{zw_{i}}=1$ . The generalized WBM (GWBM) operator is defined:

$\displaystyle\text{GWBM}_{zw}^{p,q,r}(xa_{1},xa_{2},\ldots,xa_{n})=\left(\sum% \limits_{i,j,k=1}^{n}{zw_{i}zw_{j}zw_{k}}xa_{i}^{p}xa_{j}^{q}xa_{k}^{r}\right)% ^{1/(p+q+r)}$ (5)

3. GSVNNPWBM operator

Then, Xu and Wang [75] defined the generalized single-valued neutrosophic number WBM (GSVNNWBM) operator based on the classical GWBM operator [70].

Definition 7. Let $s,t,r>0$ , $ZA_{i}=(ZT_{i},ZI_{i},ZF_{i})(i=1,2,\ldots,n)$ be a set of SVNNs with their weight $zw_{i}=(zw_{1},zw_{2},\ldots,zw_{n})^{T}$ , $\sum\nolimits_{i=1}^{n}{zw_{i}}=1$ . If

$\displaystyle\text{GSVNNWBM}_{zw}^{s,t,r}(ZA_{1},ZA_{2},\ldots,ZA_{n})=\left(% \mathop{\oplus}\limits_{i,j,k=1}^{n}zw_{i}zw_{j}zw_{k}(ZA_{i}^{s}\otimes ZA_{j% }^{t}\otimes ZA_{k}^{r})\right)^{1/(s+t+r)}\text{GSVNNWBM}_{zw}^{s,t,r}(ZA_{1}% ,ZA_{2},\ldots,ZA_{n})$ (6) $\displaystyle=\left(\mathop{\oplus}\limits_{i,j,k=1}^{n}zw_{i}zw_{j}zw_{k}(ZA_% {i}^{s}\otimes ZA_{j}^{t}\otimes ZA_{k}^{r})\right)^{1/(s+t+r)}=\left({\begin{% array}[]{l}\left(1-\mathop{\Pi}\limits_{i,j,k=1}^{n}(1-ZT_{i}^{s}ZT_{j}^{t}ZT_% {k}^{r})^{zw_{i}zw_{j}zw_{k}}\right)^{1/(s+t+r)},\\ 1-\left(1-\mathop{\Pi}\limits_{i,j,k=1}^{n}(1-(1-ZI_{i})^{s}(1-ZI_{j})^{t}(1-% ZI_{k})^{r})^{zw_{i}zw_{j}zw_{k}}\right)^{1/(s+t+r)},\\ 1-\left(1-\mathop{\Pi}\limits_{i,j,k=1}^{n}(1-(1-ZF_{i})^{s}(1-ZF_{j})^{t}(1-% ZF_{k})^{r})^{zw_{i}zw_{j}zw_{k}}\right)^{1/(s+t+r)}.\\ \end{array}}\right)$ (7)

then $\text{GSVNNWBM}_{zw}^{s,t,r}$ is named the GSVNNWBM operator.

The GSVNNWBM operator [75] has the three properties.

Property 1. (idempotency). If $ZA_{i}=ZA=(ZT,ZI,ZF)$ ,

$\displaystyle\text{GSVNNWBM}_{zw}^{s,t,r}(ZA_{1},ZA_{2},\ldots,ZA_{n})=ZA$ (8)

Property 2. (Monotonicity). Let $ZA_{i}=(TT_{A_{i}},II_{A_{i}},FF_{A_{i}})$ , $ZB_{i}=(ZT_{B_{i}},ZI_{B_{i}},ZF_{B_{i}})$ . If $ZT_{A_{i}}\leqslant ZT_{B_{i}},Z_{A_{i}}\geqslant ZI_{B_{i}},ZF_{A_{i}}% \geqslant ZF_{B_{i}}$ holds for all i, then

$\displaystyle\text{GSVNNWBM}_{zw}^{s,t,r}(ZA_{1},ZA_{2},\ldots,ZA_{n})% \leqslant\text{GSVNNWBM}_{zw}^{s,t,r}(ZB_{1},ZB_{2},\ldots,ZB_{n})$ (9)

Property 3. (Boundedness). Let $ZA_{i}=(ZT_{A_{i}},ZI_{A_{i}},ZF_{A_{i}})$ . If $ZA^{+}=(\max_{i}(TT_{i}),\min_{i}(II_{i}),\min_{i}\linebreak(FF_{i}))$ , $ZA^{-}=(\min_{i}(TT_{i}),\max_{i}(II_{i}),\max_{i}(FF_{i}))$ , then

$\displaystyle ZA^{-}\leqslant\text{GSVNNWBM}_{zw}^{s,t,r}(ZA_{1},ZA_{2},\ldots% ,ZA_{n})\leqslant ZA^{+}$ (10)

Then, the generalized single-valued neutrosophic number power WBM (GSVNNPWBM) operator is built on the GSVNNWBM operator [75] and PA operator [71].

Definition 8. Let $s,t,r>0$ , $ZA_{i}=(ZT_{i},ZI_{i},ZF_{i})(i=1,2,\ldots,n)$ be a set of SVNNs with their weight $zw_{i}=(zw_{1},zw_{2},\ldots,zw_{n})^{T}$ , $\sum\nolimits_{i=1}^{n}{zw_{i}}=1$ . If

where $T({ZA_{a}})=\sum\limits_{\begin{subarray}{c}j=1\\ a\neq j\end{subarray}}^{m}zw_{j}\textit{Sup}({ZA_{a},ZA_{j}})$ , $\textit{Sup}({ZA_{a},ZA_{j}})$ is the decision support for $ZA_{a}$ from $ZA_{j}$ , with given conditions: (1) $\textit{Sup}({ZA_{a},ZA_{b}})\in[{0,1}]$ ; (2) $\textit{Sup}({ZA_{b},ZA_{a}})=\textit{Sup}({ZA_{a},ZA_{b}})$ ; (3) $\textit{Sup}({ZA_{a},ZA_{b}})\geqslant\textit{Sup}({ZA_{s},ZA_{t}})$ , if $d({ZA_{a},ZA_{b}})\geqslant d({ZA_{s},ZA_{t}})$ , where $d$ is a distance.

The Theorem 1 is obtained.

Theorem 1. Let $s,t,r>0$ , $ZA_{i}=(ZT_{i},ZI_{i},ZF_{i})(i=1,2,\ldots,n)$ . The fused value is the SVNN and

$\displaystyle\text{GSVNNPWBM}_{zw}^{s,t,r}(ZA_{1},ZA_{2},\ldots,ZA_{n})=\left(% \begin{array}[]{l}\mathop{\oplus}\limits_{i,j,k=1}^{n}\frac{zw_{i}({1+T({ZA_{i% }})})}{\sum\limits_{i=1}^{n}{zw_{i}({1+T({ZA_{i}})})}}\frac{zw_{j}({1+T({ZA_{j% }})})}{\sum\limits_{j=1}^{n}{zw_{j}({1+T({ZA_{j}})})}}\frac{zw_{k}({1+T({ZA_{k% }})})}{\sum\limits_{k=1}^{n}{zw_{k}({1+T({ZA_{k}})})}}\\ ({ZA_{i}^{s}\otimes ZA_{j}^{t}\otimes ZA_{k}^{r}})\\ \end{array}\right)^{1/(s+t+r)}=\left(\begin{array}[]{l}\left(\begin{array}[]{l% }1-\mathop{\Pi}\limits_{i,j,k=1}^{n}({1-ZT_{i}^{s}ZT_{j}^{t}ZT_{k}^{r}})\\ \frac{zw_{i}({1+T({ZA_{i}})})}{\sum\limits_{i=1}^{n}{zw_{i}({1+T({ZA_{i}})})}}% \frac{zw_{j}({1+T({ZA_{j}})})}{\sum\limits_{j=1}^{n}{zw_{j}({1+T({ZA_{j}})})}}% \frac{zw_{k}({1+T({ZA_{k}})})}{\sum\limits_{k=1}^{n}{zw_{k}({1+T({ZA_{k}})})}}% \end{array}\right)^{1/(s+t+r)},\\ 1-\left(\begin{array}[]{l}1-\mathop{\Pi}\limits_{i,j,k=1}^{n}({1-(1-ZI_{i})^{s% }(1-ZI_{j})^{t}(1-ZI_{k})^{r}})\\ \frac{zw_{i}({1+T({ZA_{i}})})}{\sum\limits_{i=1}^{n}{zw_{i}({1+T({ZA_{i}})})}}% \frac{zw_{j}({1+T({ZA_{j}})})}{\sum\limits_{j=1}^{n}{zw_{j}({1+T({ZA_{j}})})}}% \frac{zw_{k}({1+T({ZA_{k}})})}{\sum\limits_{k=1}^{n}{zw_{k}({1+T({ZA_{k}})})}}% \end{array}\right)^{1/(s+t+r)},\\ 1-\left(\begin{array}[]{l}1-\mathop{\Pi}\limits_{i,j,k=1}^{n}({1-(1-ZF_{i})^{s% }(1-ZF_{j})^{t}(1-ZF_{k})^{r}})\\ \frac{zw_{i}({1+T({ZA_{i}})})}{\sum\limits_{i=1}^{n}{zw_{i}({1+T({ZA_{i}})})}}% \frac{zw_{j}({1+T({ZA_{j}})})}{\sum\limits_{j=1}^{n}{zw_{j}({1+T({ZA_{j}})})}}% \frac{zw_{k}({1+T({ZA_{k}})})}{\sum\limits_{k=1}^{n}{zw_{k}({1+T({ZA_{k}})})}}% \end{array}\right)^{1/(s+t+r)}\\ \end{array}\right)$ (12)

Proof:

$\displaystyle ZA_{i}^{s}=(ZT_{i}^{s},1-(1-ZI_{i})^{s},1-(1-ZF_{i})^{s});$ $\displaystyle ZA_{j}^{t}=(ZT_{j}^{t},1-(1-ZI_{j})^{t},1-(1-ZF_{j})^{t});$ (13) $\displaystyle ZA_{k}^{r}=(ZT_{k}^{r},1-(1-ZI_{k})^{r},1-(1-ZF_{k})^{r}).$

Thus,

$\displaystyle ZA_{i}^{s}\otimes ZA_{j}^{t}\otimes ZA_{k}^{r}=\left({\begin{% array}[]{l}ZT_{i}^{s}ZT_{j}^{t}ZT_{k}^{r},\\ 1-(1-ZI_{i})^{s}(1-ZI_{j})^{t}(1-ZI_{k})^{r},\\ 1-(1-ZF_{i})^{s}(1-ZF_{j})^{t}(1-ZF_{k})^{r}\\ \end{array}}\right)$ (14)

Thereafter,

$\displaystyle\frac{zw_{i}({1+T({ZA_{i}})})}{\sum\limits_{i=1}^{n}{zw_{i}({1+T(% {ZA_{i}})})}}\frac{zw_{j}({1+T({ZA_{j}})})}{\sum\limits_{j=1}^{n}{zw_{j}({1+T(% {ZA_{j}})})}}\frac{zw_{k}({1+T({ZA_{k}})})}{\sum\limits_{k=1}^{n}{zw_{k}({1+T(% {ZA_{k}})})}}({ZA_{i}^{s}\otimes ZA_{j}^{t}\otimes ZA_{k}^{r}})=\left({\begin{% array}[]{l}1-({1-ZT_{i}^{s}ZT_{j}^{t}ZT_{k}^{r}}){\frac{zw_{i}({1+T({ZA_{i}})}% )}{\sum\limits_{i=1}^{n}{zw_{i}({1+T({ZA_{i}})})}}\frac{zw_{j}({1+T({ZA_{j}})}% )}{\sum\limits_{j=1}^{n}{zw_{j}({1+T({ZA_{j}})})}}\frac{zw_{k}({1+T({ZA_{k}})}% )}{\sum\limits_{k=1}^{n}{zw_{k}({1+T({ZA_{k}})})}}},\\ \begin{array}[]{l}({1-({1-(1-ZI_{i})^{s}(1-ZI_{j})^{t}(1-ZI_{k})^{r}})})\\ {\frac{zw_{i}({1+T({ZA_{i}})})}{\sum\limits_{i=1}^{n}{zw_{i}({1+T({ZA_{i}})})}% }\frac{zw_{j}({1+T({ZA_{j}})})}{\sum\limits_{j=1}^{n}{zw_{j}({1+T({ZA_{j}})})}% }\frac{zw_{k}({1+T({ZA_{k}})})}{\sum\limits_{k=1}^{n}{zw_{k}({1+T({ZA_{k}})})}% }},\end{array}\\ \begin{array}[]{l}({1-({1-(1-ZF_{i})^{s}(1-ZF_{j})^{t}(1-ZF_{k})^{r}})})\\ {\frac{zw_{i}({1+T({ZA_{i}})})}{\sum\limits_{i=1}^{n}{zw_{i}({1+T({ZA_{i}})})}% }\frac{zw_{j}({1+T({ZA_{j}})})}{\sum\limits_{j=1}^{n}{zw_{j}({1+T({ZA_{j}})})}% }\frac{zw_{k}({1+T({ZA_{k}})})}{\sum\limits_{k=1}^{n}{zw_{k}({1+T({ZA_{k}})})}% }}\end{array}\\ \end{array}}\right)$ (15)

Furthermore,

$\displaystyle\mathop{\oplus}\limits_{i,j,k=1}^{n}\frac{zw_{i}({1+T({ZA_{i}})})% }{\sum\limits_{i=1}^{n}{zw_{i}({1+T({ZA_{i}})})}}\frac{zw_{j}({1+T({ZA_{j}})})% }{\sum\limits_{j=1}^{n}{zw_{j}({1+T({ZA_{j}})})}}\frac{zw_{k}({1+T({ZA_{k}})})% }{\sum\limits_{k=1}^{n}{zw_{k}({1+T({ZA_{k}})})}}({ZA_{i}^{s}\otimes ZA_{j}^{t% }\otimes ZA_{k}^{r}})=\left({\begin{array}[]{l}1-\mathop{\Pi}\limits_{i,j,k=1}% ^{n}({1-ZT_{i}^{s}ZT_{j}^{t}ZT_{k}^{r}}){\frac{zw_{i}({1+T({ZA_{i}})})}{\sum% \limits_{i=1}^{n}{zw_{i}({1+T({ZA_{i}})})}}\frac{zw_{j}({1+T({ZA_{j}})})}{\sum% \limits_{j=1}^{n}{zw_{j}({1+T({ZA_{j}})})}}\frac{zw_{k}({1+T({ZA_{k}})})}{\sum% \limits_{k=1}^{n}{zw_{k}({1+T({ZA_{k}})})}}},\\ \begin{array}[]{l}\mathop{\Pi}\limits_{i,j,k=1}^{n}({1-(1-ZI_{i})^{s}(1-ZI_{j}% )^{t}(1-ZI_{k})^{r}})\\ {\frac{zw_{i}({1+T({ZA_{i}})})}{\sum\limits_{i=1}^{n}{zw_{i}({1+T({ZA_{i}})})}% }\frac{zw_{j}({1+T({ZA_{j}})})}{\sum\limits_{j=1}^{n}{zw_{j}({1+T({ZA_{j}})})}% }\frac{zw_{k}({1+T({ZA_{k}})})}{\sum\limits_{k=1}^{n}{zw_{k}({1+T({ZA_{k}})})}% }},\end{array}\\ \begin{array}[]{l}\mathop{\Pi}\limits_{i,j,k=1}^{n}({1-(1-ZF_{i})^{s}(1-ZF_{j}% )^{t}(1-ZF_{k})^{r}})\\ {\frac{zw_{i}({1+T({ZA_{i}})})}{\sum\limits_{i=1}^{n}{zw_{i}({1+T({ZA_{i}})})}% }\frac{zw_{j}({1+T({ZA_{j}})})}{\sum\limits_{j=1}^{n}{zw_{j}({1+T({ZA_{j}})})}% }\frac{zw_{k}({1+T({ZA_{k}})})}{\sum\limits_{k=1}^{n}{zw_{k}({1+T({ZA_{k}})})}% }}\end{array}\\ \end{array}}\right)$ (16)

Therefore,

$\displaystyle\text{GSVNNPWBM}_{zw}^{s,t,r}(ZA_{1},ZA_{2},\ldots,ZA_{n})=\left(% {\begin{array}[]{l}\left(\begin{array}[]{l}1-\mathop{\Pi}\limits_{i,j,k=1}^{n}% ({1-ZT_{i}^{s}ZT_{j}^{t}ZT_{k}^{r}})\\ {\frac{zw_{i}({1+T({ZA_{i}})})}{\sum\limits_{i=1}^{n}{zw_{i}({1+T({ZA_{i}})})}% }\frac{zw_{j}({1+T({ZA_{j}})})}{\sum\limits_{j=1}^{n}{zw_{j}({1+T({ZA_{j}})})}% }\frac{zw_{k}({1+T({ZA_{k}})})}{\sum\limits_{k=1}^{n}{zw_{k}({1+T({ZA_{k}})})}% }}\end{array}\right)^{1/(s+t+r)},\\ 1-\left(\begin{array}[]{l}1-\mathop{\Pi}\limits_{i,j,k=1}^{n}({1-(1-ZI_{i})^{s% }(1-ZI_{j})^{t}(1-ZI_{k})^{r}})\\ {\frac{zw_{i}({1+T({ZA_{i}})})}{\sum\limits_{i=1}^{n}{zw_{i}({1+T({ZA_{i}})})}% }\frac{zw_{j}({1+T({ZA_{j}})})}{\sum\limits_{j=1}^{n}{zw_{j}({1+T({ZA_{j}})})}% }\frac{zw_{k}({1+T({ZA_{k}})})}{\sum\limits_{k=1}^{n}{zw_{k}({1+T({ZA_{k}})})}% }}\end{array}\right)^{1/(s+t+r)},\\ 1-\left(\begin{array}[]{l}1-\mathop{\Pi}\limits_{i,j,k=1}^{n}({1-(1-ZF_{i})^{s% }(1-ZF_{j})^{t}(1-ZF_{k})^{r}})\\ {\frac{zw_{i}({1+T({ZA_{i}})})}{\sum\limits_{i=1}^{n}{zw_{i}({1+T({ZA_{i}})})}% }\frac{zw_{j}({1+T({ZA_{j}})})}{\sum\limits_{j=1}^{n}{zw_{j}({1+T({ZA_{j}})})}% }\frac{zw_{k}({1+T({ZA_{k}})})}{\sum\limits_{k=1}^{n}{zw_{k}({1+T({ZA_{k}})})}% }}\end{array}\right)^{1/(s+t+r)}\\ \end{array}}\right)$ (17)

Hence, Eq. (12) is met.

We have,

$\displaystyle 0\leqslant\left(\begin{array}[]{l}1-\mathop{\Pi}\limits_{i,j,k=1% }^{n}({1-ZT_{i}^{s}ZT_{j}^{t}ZT_{k}^{r}})\\ {\frac{zw_{i}({1+T({ZA_{i}})})}{\sum\limits_{i=1}^{n}{zw_{i}({1+T({ZA_{i}})})}% }\frac{zw_{j}({1+T({ZA_{j}})})}{\sum\limits_{j=1}^{n}{zw_{j}({1+T({ZA_{j}})})}% }\frac{zw_{k}({1+T({ZA_{k}})})}{\sum\limits_{k=1}^{n}{zw_{k}({1+T({ZA_{k}})})}% }}\end{array}\right)^{1/(s+t+r)}\leqslant 1,$ $\displaystyle 0\leqslant 1-\left(\begin{array}[]{l}1-\mathop{\Pi}\limits_{i,j,% k=1}^{n}({1-(1-ZI_{i})^{s}(1-ZI_{j})^{t}(1-ZI_{k})^{r}})\\ {\frac{zw_{i}({1+T({ZA_{i}})})}{\sum\limits_{i=1}^{n}{zw_{i}({1+T({ZA_{i}})})}% }\frac{zw_{j}({1+T({ZA_{j}})})}{\sum\limits_{j=1}^{n}{zw_{j}({1+T({ZA_{j}})})}% }\frac{zw_{k}({1+T({ZA_{k}})})}{\sum\limits_{k=1}^{n}{zw_{k}({1+T({ZA_{k}})})}% }}\end{array}\right)^{1/(s+t+r)}\leqslant 1,$ (18) $\displaystyle 0\leqslant 1-\left(\begin{array}[]{l}1-\mathop{\Pi}\limits_{i,j,% k=1}^{n}({1-(1-ZF_{i})^{s}(1-ZF_{j})^{t}(1-ZF_{k})^{r}})\\ {\frac{zw_{i}({1+T({ZA_{i}})})}{\sum\limits_{i=1}^{n}{zw_{i}({1+T({ZA_{i}})})}% }\frac{zw_{j}({1+T({ZA_{j}})})}{\sum\limits_{j=1}^{n}{zw_{j}({1+T({ZA_{j}})})}% }\frac{zw_{k}({1+T({ZA_{k}})})}{\sum\limits_{k=1}^{n}{zw_{k}({1+T({ZA_{k}})})}% }}\end{array}\right)^{1/(s+t+r)}\leqslant 1.$

Thereafter,

$\displaystyle 0\leqslant\left(\begin{array}[]{l}1-\mathop{\Pi}\limits_{i,j,k=1% }^{n}({1-ZT_{i}^{s}ZT_{j}^{t}ZT_{k}^{r}})\\ {\frac{zw_{i}({1+T({ZA_{i}})})}{\sum\limits_{i=1}^{n}{zw_{i}({1+T({ZA_{i}})})}% }\frac{zw_{j}({1+T({ZA_{j}})})}{\sum\limits_{j=1}^{n}{zw_{j}({1+T({ZA_{j}})})}% }\frac{zw_{k}({1+T({ZA_{k}})})}{\sum\limits_{k=1}^{n}{zw_{k}({1+T({ZA_{k}})})}% }}\end{array}\right)^{1/(s+t+r)}$ $\displaystyle{}+1-\left(\begin{array}[]{l}1-\mathop{\Pi}\limits_{i,j,k=1}^{n}(% {1-(1-ZI_{i})^{s}(1-ZI_{j})^{t}(1-ZI_{k})^{r}})\\ {\frac{zw_{i}({1+T({ZA_{i}})})}{\sum\limits_{i=1}^{n}{zw_{i}({1+T({ZA_{i}})})}% }\frac{zw_{j}({1+T({ZA_{j}})})}{\sum\limits_{j=1}^{n}{zw_{j}({1+T({ZA_{j}})})}% }\frac{zw_{k}({1+T({ZA_{k}})})}{\sum\limits_{k=1}^{n}{zw_{k}({1+T({ZA_{k}})})}% }}\end{array}\right)^{1/(s+t+r)}$ (19) $\displaystyle{}+1-\left(\begin{array}[]{l}1-\mathop{\Pi}\limits_{i,j,k=1}^{n}(% {1-(1-ZF_{i})^{s}(1-ZF_{j})^{t}(1-ZF_{k})^{r}})\\ {\frac{zw_{i}({1+T({ZA_{i}})})}{\sum\limits_{i=1}^{n}{zw_{i}({1+T({ZA_{i}})})}% }\frac{zw_{j}({1+T({ZA_{j}})})}{\sum\limits_{j=1}^{n}{zw_{j}({1+T({ZA_{j}})})}% }\frac{zw_{k}({1+T({ZA_{k}})})}{\sum\limits_{k=1}^{n}{zw_{k}({1+T({ZA_{k}})})}% }}\end{array}\right)^{1/(s+t+r)}\leqslant 3.$

Thereby finishing this proof.

The GSVNNPWBM has the three decision properties.

Property 4. (idempotency). If $ZA_{i}=ZA=(ZT,ZI,ZF)$ ,

$\displaystyle\text{GSVNNPWBM}_{zw}^{s,t,r}(ZA_{1},ZA_{2},\ldots,ZA_{n})=ZA$ (20)

Property 6. (Monotonicity). Let $ZA_{i}=(ZT_{A_{i}},ZI_{A_{i}},ZF_{A_{i}})$ , $ZB_{i}=(ZT_{B_{i}},ZI_{B_{i}},ZF_{B_{i}})$ . If $ZT_{A_{i}}\leqslant ZT_{B_{i}},ZI_{A_{i}}\geqslant ZI_{B_{i}},ZF_{A_{i}}% \geqslant ZF_{B_{i}}$ holds for all i, then

$\displaystyle\text{GSVNNPWBM}_{zw}^{s,t,r}(ZA_{1},ZA_{2},\ldots,ZA_{n})$ (21) $\displaystyle{}\leqslant\text{GSVNNPWBM}_{zw}^{s,t,r}(ZB_{1},ZB_{2},\ldots,ZB_% {n})$

Property 6. (Boundedness). Let $ZA_{i}=(ZT_{A_{i}},ZI_{A_{i}},ZF_{A_{i}})$ . If $ZA^{+}=(\max_{i}(ZT_{i}),\min_{i}(ZI_{i}),\min_{i}\linebreak(ZF_{i}))$ , $ZA^{-}=(\min_{i}(ZT_{i}),\max_{i}(ZI_{i}),\max_{i}(ZF_{i}))$ , then

$\displaystyle ZA^{-}\leqslant\text{GSVNNPWBM}_{zw}^{s,t,r}(ZA_{1},ZA_{2},% \ldots,ZA_{n})\leqslant ZA^{+}$ (22)

4. MADM method based on GSVNNPWBM with SVNNs

Then, the GSVNNPWBM method is built for MADM under SVNSs. Let $ZX=\{{ZX_{1},ZX_{2},\ldots,ZX_{n}}\}$ be attributes, $zw=\{{zw_{1},zw_{2},\ldots,zw_{n}}\}$ be weight of $ZX_{j}$ . Let $ZP=\{{ZP_{1},ZP_{2},\ldots,ZP_{m}}\}$ be alternatives. $({zq_{ij}})_{m\times n}=({ZT_{ij},ZI_{ij},ZF_{ij}})_{m\times n}$ is the SVNNmatrix. The GSVNNPWBM operator is built to solve the MADM (See Fig. 1).

Figure 1.

The MADM method based on GSVNNPWBM with SVNNs.

Step 1. Structure SVNN matrix $\textit{ZXQ}=({zq_{ij}})_{m\times n}=({ZT_{ij},ZI_{ij},ZF_{ij}})_{m\times n}$ .

$\displaystyle ZQ=[{zq{}_{ij}}]_{m\times n}=\left[{{\begin{array}[]{*{20}c}{zq_% {11}}&{zq_{12}}&\ldots&{zq_{1n}}\\ {zq_{21}}&{zq_{22}}&\ldots&{zq_{2n}}\\ \vdots&\vdots&\vdots&\vdots\\ {zq_{m1}}&{zq_{m2}}&\ldots&{zq_{mn}}\\ \end{array}}}\right]$ (23) $\displaystyle ZQ=[{zq_{ij}}]_{m\times n}=({ZT_{ij},ZI_{ij},ZF_{ij}})_{m\times n% }=\left[{{\begin{array}[]{*{20}c}{({ZT_{11},ZI_{11},ZF_{11}})}&{({ZT_{12},ZI_{% 1},ZF_{12}})}&\ldots&{({ZT_{1n},ZI_{1n},ZF_{1n}})}\\ {({ZT_{21},ZI_{21},ZF_{21}})}&{({ZT_{22},ZI_{22},ZF_{22}})}&\ldots&{({ZT_{2n},% ZI_{2n},ZF_{2n}})}\\ \vdots&\vdots&\vdots&\vdots\\ {({ZT_{m1},ZI_{m1},ZF_{m1}})}&{({ZT_{m2},ZI_{m2},ZF_{m2}})}&\ldots&{({ZT_{mn},% ZI_{mn},ZF_{mn}})}\\ \end{array}}}\right]$ (24)

Step 2. Normalize the $ZQ=[{zq_{ij}}]_{m\times n}$ to $\textit{NZQ}=[{\textit{nzq}_{ij}}]_{m\times n}$ .

$\displaystyle\textit{nzq}_{ij}=({\textit{NZT}_{ij},\textit{NZI}_{ij},\textit{% NZF}_{ij}})=\left\{{{\begin{array}[]{l}({ZT_{ij},ZI_{ij},ZF_{ij}}),XZ_{j}% \textit{ is a benefit criterion}\\ ({ZF_{ij},1-ZI_{ij},ZT_{ij}}),XZ_{j}\textit{ is a cost criterion}\\ \end{array}}}\right.$ (25)

Step 3. Utilize the maximizing deviation method to determine the weight of attributes.

The maximizing deviation method [72] is employed to determine the weight under SVNSs. The procedures will be presented.

(1) Depending on $\textit{NZQ}=[{\textit{nzq}_{ij}}]_{m\times n}$ , the deviation $ZP_{i}$ to all other alternatives is calculated.

$\displaystyle DM_{ij}=\sum\limits_{t=1}^{m}zw_{j}\cdot DM({\textit{nzq}_{ij},% \textit{nzq}_{tj}})$ (26)

where

$\displaystyle DM({\textit{nzq}_{ij},\textit{nzq}_{tj}})=\frac{|{ZT_{ij}-ZT_{tj% }+ZI_{ij}-ZI_{tj}}|}{4}+\frac{|{ZT_{ij}-ZT_{tj}+FI_{ij}-FI_{tj}}|}{4}.$

(2) Structure the total weighted deviation.

$\displaystyle DM({wz})=\sum\limits_{i=1}^{m}{\textit{SVNNHD}_{ij}}({wz})=\sum% \limits_{i=1}^{m}\sum\limits_{t=1}^{m}{zw_{j}}\left({\frac{|{ZT_{ij}-ZT_{tj}+% ZI_{ij}-ZI_{tj}}|}{4}+\frac{|{ZT_{ij}-ZT_{tj}+FI_{ij}-FI_{tj}}|}{4}}\right)$ (27)

(3) Structure non-linear programming method with SVNNs.

$\displaystyle({M-1})\left\{{\begin{array}[]{l}\max DM({zw})=\sum\limits_{j=1}^% {n}{\sum\limits_{i=1}^{m}{\sum\limits_{t=1}^{m}{\mbox{zw}_{j}}}\left({\frac{|{% ZT_{ij}-ZT_{tj}+ZI_{ij}-ZI_{tj}}|}{4}+\frac{|{ZT_{ij}-ZT_{tj}+FI_{ij}-FI_{tj}}% |}{4}}\right)}\\ s.t.\,zw_{j}\geqslant 0,\sum\limits_{j=1}^{n}{zw_{j}^{2}=1}\\ \end{array}}\right.$ (28)

The Lagrange function is utilized to solve this defined model.

$\displaystyle L({zw,\xi})=\sum\limits_{j=1}^{n}\sum\limits_{i=1}^{m}{\sum% \limits_{t=1}^{m}{zw_{j}}}\left({\frac{|{ZT_{ij}-ZT_{tj}+ZI_{ij}-ZI_{tj}}|}{4}% +\frac{|{ZT_{ij}-ZT_{tj}+FI_{ij}-FI_{tj}}|}{4}}\right){}+\frac{\xi}{2}\left({% \sum\limits_{j=1}^{n}{zw_{j}^{2}-1}}\right)$

where $\xi$ is the defined Lagrange multiplier. Then the partial derivatives is calculated.

$\displaystyle\left\{{\begin{array}[]{l}\frac{\partial L}{\partial zw_{j}}=\sum% \limits_{i=1}^{m}\sum\limits_{t=1}^{m}\left({\frac{|{ZT_{ij}-ZT_{tj}+ZI_{ij}-% ZI_{tj}}|}{4}+\frac{|{ZT_{ij}-ZT_{tj}+FI_{ij}-FI_{tj}}|}{4}}\right)+\xi zw_{j}% =0\\ \frac{\partial L}{\partial\xi}=\frac{1}{2}\left({\sum\limits_{j=1}^{n}{zw_{j}^% {2}-1}}\right)=0\\ \end{array}}\right.$

The weight is obtained:

$\displaystyle zw_{j}^{*}=\frac{\sum\limits_{i=1}^{m}{\sum\limits_{t=1}^{m}{% \left({\frac{|{ZT_{ij}-ZT_{tj}+ZI_{ij}-ZI_{tj}}|}{4}+\frac{|{ZT_{ij}-ZT_{tj}+% FI_{ij}-FI_{tj}}|}{4}}\right)}}}{\sqrt{\sum\limits_{j=1}^{n}{\left({\sum% \limits_{i=1}^{m}{\sum\limits_{t=1}^{m}{\left({\frac{|{ZT_{ij}-ZT_{tj}+ZI_{ij}% -ZI_{tj}}|}{4}+\frac{|{ZT_{ij}-ZT_{tj}+FI_{ij}-FI_{tj}}|}{4}}\right)}}}\right)% ^{2}}}}$

Finally, the given normalized weights are:

$\displaystyle zw_{j}=\frac{\sum\limits_{i=1}^{m}{\sum\limits_{t=1}^{m}{\left({% \frac{|{ZT_{ij}-ZT_{tj}+ZI_{ij}-ZI_{tj}}|}{4}+\frac{|{ZT_{ij}-ZT_{tj}+FI_{ij}-% FI_{tj}}|}{4}}\right)}}}{\sum\limits_{j=1}^{n}{\sum\limits_{i=1}^{m}{\sum% \limits_{t=1}^{m}{\left({\frac{|{ZT_{ij}-ZT_{tj}+ZI_{ij}-ZI_{tj}}|}{4}+\frac{|% {ZT_{ij}-ZT_{tj}+FI_{ij}-FI_{tj}}|}{4}}\right)}}}}$ (29)

Step 4. According to $\textit{NZQ}=[{\textit{nzq}_{ij}}]_{m\times n}$ , we can fuse all SVNNs $\textit{NZQ}=[{\textit{nzq}_{ij}}]_{m\times n}$ through GSVNNPWBM operator to get the SVNNs $\textit{nzq}_{i}({i=1,2,\ldots,m})$ :

$\displaystyle zq_{i}=\text{GSVNNPWBM}_{zw}^{s,t,r}(\textit{nzq}_{i1},\textit{% nzq}_{i2},\ldots,\textit{nzq}_{in})=\left({\begin{array}[]{l}\mathop{\oplus}% \limits_{i,j,k=1}^{n}\frac{zw_{i}({1+T({\textit{nzq}_{if}})})}{\sum\limits_{i=% 1}^{n}{zw_{i}({1+T({\textit{nzq}_{if}})})}}\frac{zw_{j}({1+T({\textit{nzq}_{ij% }})})}{\sum\limits_{j=1}^{n}{zw_{j}({1+T({\textit{nzq}_{ij}})})}}\cdot\\ \frac{zw_{k}({1+T({\textit{nzq}_{ik}})})}{\sum\limits_{k=1}^{n}{zw_{k}({1+T({% \textit{nzzq}_{ik}})})}}({\textit{nzq}_{if}^{s}\otimes\textit{nzq}_{ij}^{t}% \otimes\textit{nzq}_{ik}^{r}})\\ \end{array}}\right)^{1/(s+t+r)}=\left(\begin{array}[]{l}\left(\begin{array}[]{% l}1-\mathop{\Pi}\limits_{i,j,k=1}^{n}({1-\textit{NZT}_{if}^{s}\textit{NZT}_{ij% }^{t}\textit{NZT}_{ik}^{r}})\\ {\frac{zw_{i}({1+T({\textit{nzq}_{if}})})}{\sum\limits_{i=1}^{n}{zw_{i}({1+T({% \textit{nzq}_{if}})})}}\frac{zw_{j}({1+T({\textit{nzq}_{ij}})})}{\sum\limits_{% j=1}^{n}{zw_{j}({1+T({\textit{nzq}_{ij}})})}}\frac{zw_{k}({1+T({\textit{nzq}_{% ik}})})}{\sum\limits_{k=1}^{n}{zw_{k}({1+T({\textit{nzzq}_{ik}})})}}}\end{% array}\right)^{1/(s+t+r)},\\ \begin{array}[]{l}({1-({1-(1-\textit{NZI}_{if})^{s}(1-\textit{NZI}_{ij})^{t}(1% -\textit{NZI}_{ik})^{r}})})\\ \frac{zw_{i}({1+T({\textit{nzq}_{if}})})}{\sum\limits_{i=1}^{n}{zw_{i}({1+T({% \textit{nzq}_{if}})})}}\frac{zw_{j}({1+T({\textit{nzq}_{ij}})})}{\sum\limits_{% j=1}^{n}{zw_{j}({1+T({\textit{nzq}_{ij}})})}}\frac{zw_{k}({1+T({\textit{nzq}_{% ik}})})}{\sum\limits_{k=1}^{n}{zw_{k}({1+T({\textit{nzzq}_{ik}})})}},\end{% array}\\ \begin{array}[]{l}({1-({1-(1-\textit{NZF}_{if})^{s}(1-\textit{NZF}_{ij})^{t}(1% -\textit{NZF}_{ik})^{r}})})\\ \frac{zw_{i}({1+T({\textit{nzq}_{if}})})}{\sum\limits_{i=1}^{n}{zw_{i}({1+T({% \textit{nzq}_{if}})})}}\frac{zw_{j}({1+T({\textit{nzq}_{ij}})})}{\sum\limits_{% j=1}^{n}{zw_{j}({1+T({\textit{nzq}_{ij}})})}}\frac{zw_{k}({1+T({\textit{nzq}_{% ik}})})}{\sum\limits_{k=1}^{n}{zw_{k}({1+T({\textit{nzzq}_{ik}})})}}\end{array% }\end{array}\right)$ (30)

Step 5. Obtain the $SV({\textit{nzq}_{i}})$ , $AV({\textit{nzq}_{i}})({i=1,2,\ldots,m})$ .

$\displaystyle SV({\textit{nzq}_{i}})=\frac{({2+\textit{NZT}_{i}-\textit{NZI}_{% i}-\textit{NZF}_{i}})}{3},AV({\textit{nzq}_{i}})=\textit{NZT}_{i}-\textit{NZF}% _{i}$ (31)

Step 6. Rank the $ZP_{i}({i=1,2,\ldots,m})$ through $SV({\textit{nzq}_{i}})$ , $AV({\textit{nzq}_{i}})$ .

5. An empirical example and comparative analysis

In this section, the GSVNNPWBM operator is used to solve the chair furniture comfort design evaluation. Then, an example about chair furniture comfort design evaluation and influence analysis and some comparative analysis were given to demonstrate the GSVNNPWBM operator.

5.1 An empirical example

As furniture itself, its narrow definition refers to the appliances used in indoor life, which are necessary facilities to generate specific practical value in the building space; In a broad sense, furniture is an indispensable tool for people’s daily life, work, and social activities, as well as a common product aimed at meeting daily needs and pursuing visual expression and ideals. Therefore, it can be considered that the broad concept of public seats is a tool used for public activities of the crowd to meet the sitting and leaning functions. It is the material foundation that determines the function of public space and an important element that represents the form of public space. Public seats belong to a type of public furniture, and the distinguishing feature of public seats from general seats is that they have the characteristics of “commonality” and “communication” in a universal sense. According to the usage situation, it can be divided into public indoor furniture and public outdoor furniture. Public indoor furniture can be divided into school furniture, commercial furniture, cinema furniture, transportation furniture, etc. Modern life respects individual choices, while new ones arise from boredom with the old. With the continuous improvement of material quality of life, people are increasingly pursuing romantic, exquisite, practical, and high-quality social activity environments. People begin to hold great aspirations for spiritual life, and are more interested in their own lives, success, and money. They actively enjoy life and pursue spiritual satisfaction. As a result, activities in public places are becoming increasingly frequent. As an important component of public space and a leisure “auxiliary tool” – public seats, their design and research have also received increasing attention and attention from people. The demand for such furniture in China is increasing day by day, but the design and research work of public seats lags behind the development status and speed of other civil and office furniture. It is particularly urgent to understand the role and impact of public seats on the overall indoor environment, as well as the new design concept of public seats established in the context of the overall environmental system. The chair furniture comfort design evaluation is a classical MADM issue. In this paper, an empirical application of chair furniture comfort design evaluation is given through GSVNNPWBM method. There are five chair furniture design schemes are evaluated their comfort design quality. In order to assess five chair furniture design schemes fairly, the experts give their information with the four attributes (Table 1).

Table 1
The given four attributes

Attributes	Description
XZ ${}_{1}$	Comfort usage perception of chair furniture
XZ ${}_{2}$	Comfort manufacturing cost of chair furniture
XZ ${}_{3}$	Comfort visual perception of chair furniture
XZ ${}_{4}$	Comfortable emotional experience of chair furniture

Evidently, XZ ${}_{2}$ is the cost, others are the benefit. Then, the GSVNNPWBM method is applied to MADM for solving the chair furniture comfort design evaluation with SVNNs. The GSVNNPWBM method involves the decision steps as below:

Step 1. Set up the SVNN-matrix $ZQ=({zq_{ij}})_{5\times 4}$ as in Table 2.

Table 2

SVNN matrix

	ZX ${}_{1}$	ZX ${}_{2}$	ZX ${}_{3}$	ZX ${}_{4}$
ZP ${}_{1}$	(0.53, 0.29, 0.58)	(0.57, 0.36, 0.32)	(0.53, 0.45, 0.37)	(0.45, 0.34, 0.62)
ZP ${}_{2}$	(0.48, 0.26, 0.52)	(0.54, 0.27, 0.53)	(0.56, 0.21, 0.52)	(0.37, 0.28, 0.54)
ZP ${}_{3}$	(0.39, 0.23, 0.34)	(0.24, 0.29, 0.26)	(0.32, 0.54, 0.34)	(0.23, 0.32, 0.46)
ZP ${}_{4}$	(0.34, 0.16, 0.46)	(0.48, 0.25, 0.32)	(0.28, 0.34, 0.65)	(0.37, 0.48, 0.36)
ZP ${}_{5}$	(0.57, 0.34, 0.53)	(0.56, 0.24, 0.63)	(0.36, 0.28, 0.53)	(0.56, 0.17, 0.53)

Step 2. Normalize $ZQ=[{zq_{ij}}]_{5\times 4}$ to $\textit{NZQ}=[{\textit{nzq}_{ij}}]_{5\times 4}$ (See the Table 3).

Table 3

The NZQ matrix

	ZX ${}_{1}$	ZX ${}_{2}$	ZX ${}_{3}$	ZX ${}_{4}$
ZP ${}_{1}$	(0.53, 0.29, 0.58)	(0.32, 0.64, 0.57)	(0.53, 0.45, 0.37)	(0.45, 0.34, 0.62)
ZP ${}_{2}$	(0.48, 0.26, 0.52)	(0.53, 0.73, 0.54)	(0.56, 0.21, 0.52)	(0.37, 0.28, 0.54)
ZP ${}_{3}$	(0.39, 0.23, 0.34)	(0.26, 0.71, 0.24)	(0.32, 0.54, 0.34)	(0.23, 0.32, 0.46)
ZP ${}_{4}$	(0.34, 0.16, 0.46)	(0.32, 0.75, 0.48)	(0.28, 0.34, 0.65)	(0.37, 0.48, 0.36)
ZP ${}_{5}$	(0.57, 0.34, 0.53)	(0.63, 0.76, 0.56)	(0.36, 0.28, 0.53)	(0.56, 0.17, 0.53)

Step 3. The attribute weights values were constructed in Table 4.

Table 4

The weight information

	ZX ${}_{1}$	ZX ${}_{2}$	ZX ${}_{3}$	ZX ${}_{4}$
$z w$	0.2289	0.1391	0.2953	0.3367

Table 5

The $\textit{nzq}_{i}({i=1,2,\ldots,5})$ by GSVNNPWBM operator ( $s=t=r=1$ )

Alternatives	$\textit{nzq}_{i}({i=1,2,\ldots,5})$
ZP ${}_{1}$	(0.5226, 0.4594, 0.3458)
ZP ${}_{2}$	(0.5875, 0.5235, 0.1847)
ZP ${}_{3}$	(0.5304, 0.4267, 0.3193)
ZP ${}_{4}$	(0.6643, 0.2732, 0.3254)
ZP ${}_{5}$	(0.5216, 0.3403, 0.2896)

Table 6

The $SV({\textit{nzq}_{i}})({i=1,2,\ldots,5})$

Alternatives	$SV({\textit{nzq}_{i}})({i=1,2,\ldots,5})$	Order
ZP ${}_{1}$	0.5725	5
ZP ${}_{2}$	0.6264	3
ZP ${}_{3}$	0.5948	4
ZP ${}_{4}$	0.6886	1
ZP ${}_{5}$	0.6320	2

Table 7

Different decision parameters for GSVNNWBM

( $s, t, r$ )	$SV({ZP_{1}})$	$SV({ZP_{2}})$	$SV({ZP_{3}})$	$SV({ZP_{4}})$	$SV({ZP_{5}})$
(1, 1, 1)	0.5725	0.6264	0.5948	0.6886	0.6320
(2, 2, 2)	0.5899	0.6438	0.6122	0.7060	0.6494
(3, 3, 3)	0.6072	0.6611	0.6295	0.7233	0.6667
(4, 4, 4)	0.6246	0.6785	0.6469	0.7407	0.6841
(5, 5, 5)	0.6419	0.6958	0.6642	0.7580	0.7014
(6, 6, 6)	0.6593	0.7132	0.6816	0.7754	0.7188
(7, 7, 7)	0.6767	0.7306	0.6990	0.7928	0.7362
(8, 8, 8)	0.6940	0.7479	0.7163	0.8101	0.7535
(9, 9, 9)	0.7114	0.7653	0.7337	0.8275	0.7709
(10, 10, 10)	0.7287	0.7826	0.7510	0.8448	0.7882

Table 8

Different order for GSVNNPWBM

( $s, t, r$ )	Order
(1, 1, 1)	$ZP_{4}>ZP_{5}>ZP_{2}>ZP_{3}>ZP_{1}$
(2, 2, 2)	$ZP_{4}>ZP_{5}>ZP_{2}>ZP_{3}>ZP_{1}$
(3, 3, 3)	$ZP_{4}>ZP_{5}>ZP_{2}>ZP_{3}>ZP_{1}$
(4, 4, 4)	$ZP_{4}>ZP_{5}>ZP_{2}>ZP_{3}>ZP_{1}$
(5, 5, 5)	$ZP_{4}>ZP_{5}>ZP_{2}>ZP_{3}>ZP_{1}$
(6, 6, 6)	$ZP_{4}>ZP_{5}>ZP_{2}>ZP_{3}>ZP_{1}$
(7, 7, 7)	$ZP_{4}>ZP_{5}>ZP_{2}>ZP_{3}>ZP_{1}$
(8, 8, 8)	$ZP_{4}>ZP_{5}>ZP_{2}>ZP_{3}>ZP_{1}$
(9, 9, 9)	$ZP_{4}>ZP_{5}>ZP_{2}>ZP_{3}>ZP_{1}$
(10, 10, 10)	$ZP_{4}>ZP_{5}>ZP_{2}>ZP_{3}>ZP_{1}$

Step 4. Obtain the $\textit{nzq}_{i}({i=1,2,\ldots,5})$ by utilizing GSVNNPWBM operator (Table 5).

Step 5. Obtain the $SV({\textit{nzq}_{i}})({i=1,2,\ldots,5})$ (Table 6).

Step 6. From Table 5, the order is $ZP_{4}>ZP_{5}>ZP_{2}>ZP_{3}>ZP_{1}$ , and the best chair furniture design scheme is $ZP_{4}$ .

5.2 Influence analysis

To show the effects on the ranking decision results through different decision parameters of GSVNNPWBM, the obtained results are in Tables 7 and 8.

It can be seen from the decision information in Tables 7 and 8 that when different parameter values are used, the priority of advantages and disadvantages is slightly different. In the decision-making process, the choice of parameter values can be changed according to the subjective attitude of the decision-maker.

5.3 Comparative analysis

The GSVNNPWBM operator is made comparison with SVNNWA and SVNNWG operator [73], SVNN-CODAS method [76] and SVNN-EDAS method [77]. The results are recorded in Table 9.

Table 9
Results of different methods

Methods	Order	The optimal choice
SVNNWA operator [73]	$ZP_{4}>ZP_{5}>ZP_{2}>ZP_{3}>ZP_{1}$	$ZP_{4}$
SVNNWG operator [73]	$ZP_{4}>ZP_{5}>ZP_{3}>ZP_{2}>ZP_{1}$	$ZP_{4}$
SVNN-CODAS method [76]	$ZP_{4}>ZP_{5}>ZP_{2}>ZP_{3}>ZP_{1}$	$ZP_{4}$
SVNN-EDAS method [77]	$ZP_{4}>ZP_{5}>ZP_{2}>ZP_{3}>ZP_{1}$	$ZP_{4}$
GSVNNPWBM operator	$ZP_{4}>ZP_{5}>ZP_{2}>ZP_{3}>ZP_{1}$	$ZP_{4}$

Obtained from Table 9, it is evident that the given optimal chair furniture design scheme is $ZP_{4}$ , while the worst chair furniture design scheme is $ZP_{1}$ . In other words, the ranking results of these five methods are slightly different. Different methods can effectively solve the MADM problem from different angles. The proposed GSVNNPWBM can effectively capture the intrinsic connection between attributes in MADM problems and the decision maker can change the parameters in the operator according to his own decision preference, risk attitude and other subjective consciousness in the decision process, so that the decision result can reach the satisfaction of the decision maker.

6. Conclusion

Chair furniture is an important component of furniture products, and its design needs to balance comfort and aesthetics. Sitting comfort is the core content of chair furniture design. The seat transfers the mass of the upper body of the human body to the ground through its functional parts (seat surface, backrest, and armrest), effectively reducing the load of lower body strength, reducing blood circulation in the lower limbs, and reducing fatigue caused by organic consumption, thereby playing a role in body support, assisting work and learning, and protecting physical health. Chair furniture mainly conducts comfort research using office seats, transportation seats, and student seats as carriers, emphasizing the physiological reactions of the human body during the sitting process, and conducting comprehensive evaluations based on the subjective psychological reactions of users. However, in the product development process, enterprises are mainly affected by cost, market, technology, development cycle, and other factors, and mainly test mechanical properties. It is not convenient to use scientific research equipment for comfort evaluation and research. The chair furniture comfort design evaluation is a classical MADM issues. In such paper, the generalized WBM operator is designed for solving the MADM under SVNSs. The GSVNNPWBM operator is built and then the MADM methods are proposed based on the GSVNNPWBM operator. Eventually, an example about development level evaluation of rural preschool education and some comparative decision analysis are given to demonstrate the GSVNNPWBM method. The main contribution of this paper is constructed: (1) The maximizing deviation method is employed to determine the weight under SVNSs; (2) the generalized weighted Bonferroni mean (WBM) operator and power average (PA) is extended to SVNSs; (3) the generalized single-valued neutrosophic number power WBM (GSVNNPWBM) operator is built to solve the MADM issue; (4) Finally, an example about chair furniture comfort design evaluation and some comparative analysis were given to demonstrate the GSVNNPWBM operator.

The research in this paper is part of the research on the chair furniture comfort design evaluation and is far from being able to evaluate the chair furniture comfort design evaluation in general chair furniture, there may be some possible limitations of this research, which can be further explored in future research: (1) It is a worthwhile research topic to apply prospect theory [78, 79] to MAGDM under SVNSs; (2) It is also worthwhile to apply regret theory [80] to the study of MAGDM under SVNSs. (3) It is also worthwhile to some other determining criterion weight coefficients, such as, Defining Interrelationships Between Ranked criteria (DIBR) [81, 82], Full Consistency Method (FUCOM) [83] and Logarithm Methodology of Additive Weights (LMAW) [84].

References

Gustafsson

. Furniture design by use of the finite-element method. Holz Als Roh-Und Werkst. 1995; 53(4): 257-60.

Knight

Noyes

. Children’s behaviour and the design of school furniture. Ergonomics. 1999; 42(5): 747-60.

Domljan

Grbac

, Zu, editors. Ergonomic principles relating to the design of school furniture. International Conference on Furniture Industry Adjustment to European Standards; 2003 Oct 17; Zagreb, CROATIA. ZAGREB: Univ Zagreb, Fac Forestry; 2003.

Vlaovic

Domljan

Horvat

, Zu, editors. Design and construction of the office and school furniture according to the european standards. International Conference on Furniture Industry Adjustment to European Standards; 2003 Oct 17; Zagreb, CROATIA. ZAGREB: Univ Zagreb, Fac Forestry; 2003.

Chung

JWY

Wong

TKS

. Anthropometric evaluation for primary school furniture design. Ergonomics. 2007; 50(3): 323-34.

Domljan

Grbac

Hadina

. Classroom furniture design – Correlation of pupil and chair dimensions. Coll Anthropol. 2008; 32(1): 257-65.

Tunay

Melemez

. An analysis of biomechanical and anthropometric parameters on classroom furniture design. Afr J Biotechnol. 2008; 7(8): 1081-6.

Vlaovic

Bogner

Grbac

. Comfort evaluation as the example of anthropotechnical furniture design. Coll Anthropol. 2008; 32(1): 277-83.

Mokdad

Al-Ansari

. Anthropometrics for the design of Bahraini school furniture. International Journal of Industrial Ergonomics. 2009; 39(5): 728-35.

10.

Smardzewski

. Antropotechnical aspects of furniture design. Drv Ind. 2009; 60(1): 15-21.

11.

Sutcu

Tanritanir

Durmusoglu

Koruca

. An integrated methodology for layout design and work organisation in a furniture manufacturing plant. South African Journal of Industrial Engineering. 2011; 22(1): 183-97.

12.

Abdullah

MFA

Zahari

Lamat

, editors. Industrial Design Innovation of Sarawak Contemporary Furniture Design. Malaysian-Technical-Universities Conference on Engineering and Technology (MUCET); 2012 Nov 11–12; Malaysia. AMSTERDAM: Elsevier Science Bv; 2013.

13.

Agha

Alnahhal

. Neural network and multiple linear regression to predict school children dimensions for ergonomic school furniture design. Applied Ergonomics. 2012; 43(6): 979-84.

14.

Domljan

Vlaovic

Grbac

Jajcinovic

, editors. New approaches and concepts in designing contemporary school furniture. 23rd international scientific conference on wood is good – with knowledge and technology to a competitive forestry and wood technology sector; 2012 Oct 12; Zagreb, CROATIA. ZAGREB: Zagreb Univ, Fac Forestry; 2012.

15.

Goncalves

Arezes

. Postural assessment of school children: An input for the design of furniture. Work-a Journal of Prevention Assessment & Rehabilitation. 2012; 41: 876-80.

16.

Costa

Prendeville

Beverley

Teso

Brooker

, editors. Sustainable product-service systems for an office furniture manufacturer: How insights from a pilot study can inform PSS design. 7th Industrial Product-Service Systems Conference – PSS, Industry Transformation for Sustainability and Business; 2014 May 21–22; Saint Etienne, FRANCE. AMSTERDAM: Elsevier Science Bv; 2015.

17.

Xue

Zhao

, editors. Modifying Ways of Modern Local Furniture Design to The Ming-style Circle Chairs. 3rd International Conference on Manufacturing and Industrial Technologies; 2016 May 25–27; Istanbul, TURKEY. CEDEX A: E D P Sciences; 2016.

18.

Aguilar

CMG

Panameno

Velazquez

Alvarez

BEA

Kiperstok

Cesar

. Cleaner Production Applied in a Small Furniture Industry in Brazil: Addressing Focused Changes in Design to Reduce Waste. Sustainability. 2017; 9(10): 17.

19.

Gao

Destech Publicat

, editors. Modern Chinese Wooden Chair Design Under Ming Style Furniture Neo-Confucianism. International Conference on Information, Computer and Education Engineering (ICICEE); 2017 Nov 11–12; Hong Kong, HONG KONG. LANCASTER: Destech Publications, Inc; 2017.

20.

Karageorgos

Rapti

Aimpanis

Birbilis

Ntintakis

Ntalos

, editors. Intelligent Recommendation System for Customized Ergonomic Furniture Design. 23rd International Conference on Engineering, Technology and Innovation (ICE/ITMC); 2017 Jun 27–29; Portugal. NEW YORK: Ieee; 2017.

21.

Salvador

, editor. Textile elements in a design project of children’s furniture. 1st Textile Design International Conference on Textiles, Identity and Innovation – Design the Future (D_TEX); 2017 Nov 02–04; Univ Lisboa, Lisbon Sch Architecture, Lisbon, PORTUGAL. BOCA RATON: Crc Press-Taylor & Francis Group; 2019.

22.

Shan

Jing

Effendi

MSM

Rosli

, editors. Anthropometric Evaluation and Recommendation for Primary Schools Classroom Furniture Design in Perlis. 3rd Electronic and Green Materials International Conference (EGM); 2017 Apr 29–30; Aonang Krabi, THAILAND. MELVILLE: Amer Inst Physics; 2017.

23.

Simek

, editor. Furniture testing for higher competitiveness, better quality and design. 10th Annual International Scientific Conference on More Wood, Better Management, Increasing Effectiveness: Setting Points and Perspectives; 2017 May 24–26; Prague, CZECH REPUBLIC. PRAGUE 6: Czech University Life Sciences Prague; 2017.

24.

Salvador

, editor. Human Interaction, Emotion and Sustainability: Designing Wooden Children’s Furniture. AHFE International Conference on Ergonomics in Design; 2018 Jul 21–25; Orlando, FL. CHAM: Springer International Publishing Ag; 2019.

25.

Salvador

, editor. Adapting Furniture to the Child-Ergonomics as a Main Tool in a Design Project. 20th Congress of the International-Ergonomics-Association (IEA); 2018 Aug 26–30; Florence, ITALY. CHAM: Springer International Publishing Ag; 2019.

26.

Estrada

Wyller

Dahy

, editors. Aerochair Integrative design methodologies for lightweight carbon fiber furniture design. 37th Conference on Education-and-Research-in-Computer-Aided-Architectural-Design-in-Europe (eCAADe)/23rd Conference of the Iberoamerican-Society-Digital-Graphics (SIGraDi); 2019 Sep 11–13; Univ Porto, Fac Architecture, Porto, PORTUGAL. BRUSSELS: Ecaade-Education & Research Computer Aided Architectural Design Europe; 2019.

27.

Simek

Fictum

, editors. Consequences of digitalisation on office furniture design. 12th WoodEMA Annual International Scientific Conference on Digitalisation and Circular Economy: Forestry and Forestry Based Industry Implications; 2019 Sep 11–13; Int House Sci Frederic Joliot Currie, Varna, BULGARIA. ZAGREB: Woodema, Ia-Int Assoc Econ & Manag Wood Processing & Furn Manuf; 2019.

28.

Chandra

Tag

Peiris

Minamizawa

, Ieee, editors. Preliminary Investigation of Across-Body Vibrotactile Pattern for the Design of Affective Furniture. IEEE Haptics Symposium (HAPTICS); 2020 Mar 28–31; Arlington, VA. NEW YORK: Ieee; 2020.

29.

Fabisiak

Jankowska

Klos

Knudsen

Merilampi

Priedulena

. Comparative study on design and functionality requirements for senior-friendly furniture for sitting. BioResources. 2021; 16(3): 6244-66.

30.

Incekara

. Post-COVID-19 ergonomic school furniture design under fuzzy logic. Work-a Journal of Prevention Assessment & Rehabilitation. 2021; 69(4): 1197-208.

31.

Uysal

Haviarova

. Evaluating design of mortise and tenon furniture joints under bending loads by lower tolerance limits. Wood Fiber Sci. 2021; 53(2): 109-25.

32.

Lopes

Filgueiras

Guerreiro

Monteiro

, editors. Inclusive Design: Furniture Design for Autism Parents Support. 11th International Conference on Design, User Experience, and Usability (DUXU) Held as Part of the 24th International Conference on Human-Computer Interaction (HCII); 2022 Jun 26–Jul 01; Electr Network. CHAM: Springer International Publishing Ag; 2022.

33.

Puška

Štilić

Stojanović

. Approach for multi-criteria ranking of Balkan countries based on the index of economic freedom. Journal of Decision Analytics and Intelligent Computing. 2023; 3(1): 1-14.

34.

Durmić

Stević

Chatterjee

Vasiljević

Tomašević

. Sustainable supplier selection using combined FUCOM-Rough SAW model. Reports in mechanical engineering. 2020; 1(1): 34-43.

35.

Gorcun

Senthil

Küçükönder

. Evaluation of tanker vehicle selection using a novel hybrid fuzzy MCDM technique. Decision Making: Applications in Management and Engineering. 2021; 4(2): 140-62.

36.

Bakir

Akan

Ozdemir

. Regional aircraft selection with fuzzy PIPRECIA and fuzzy MARCOS: A case study of the turkish airline industry. Facta Universitatis-Series Mechanical Engineering. 2021; 19(3): 423-45.

37.

Pamuar

Savin

. Multiple-criteria model for optimal off-road vehicle selection for passenger transportation: BWM-COPRAS model. Military Technical Courier. 2020; 68(1): 28-64.

38.

Baydas

Pamucar

. Determining Objective Characteristics of MCDM Methods under Uncertainty: An Exploration Study with Financial Data. Mathematics. 2022; 10(7).

39.

Baydas

Elma

Pamucar

. Exploring the specific capacity of different multi criteria decision making approaches under uncertainty using data from financial markets. Expert Systems with Applications. 2022; 197.

40.

da Silva

Bellinello

de Souza

GFM

Antomarioni

Bevilacqua

Ciarapica

. Deciding a Multicriteria Decision-Making (MCDM) Method to Prioritize Maintenance Work Orders of Hydroelectric Power Plants. Energies. 2021; 14(24).

41.

Anysz

Nical

Stevic

Grzegorzewski

Sikora

. Pareto optimal decisions in multi-criteria decision making explained with construction cost cases. Symmetry-Basel. 2021; 13(1).

42.

Faizi

Salabun

Rashid

Zafar

Watrobski

. Intuitionistic fuzzy sets in multi-criteria group decision making problems using the characteristic objects method. Symmetry-Basel. 2020; 12(9): 15.

43.

Watrobski

Salabun

, editors. The Characteristic Objects Method: A New Intelligent Decision Support Tool for Sustainable Manufacturing. 3rd International Conference on Sustainable Design and Manufacturing (SDM); 2016 2016 Apr 04–06; Chania, GREECE2016.

44.

Watrobski

Baczkiewicz

Ziemba

Salabun

. Sustainable cities and communities assessment using the DARIA-TOPSIS method. Sustainable Cities and Society. 2022; 83.

45.

Wieckowski

Kizielewicz

Shekhovtsov

Salabun

. RANCOM: A novel approach to identifying criteria relevance based on inaccuracy expert judgments. Engineering Applications of Artificial Intelligence. 2023; 122: 21.

46.

Yahya

Abdullah

Chinram

Al-Otaibi

Naeem

. Frank aggregation operators and their application to probabilistic hesitant fuzzy multiple attribute decision-making. International Journal of Fuzzy Systems. 2021; 23(1): 194-215.

47.

Sun

Yang

Cai

Wei

. An extended Exp-TODIM method for multiple attribute decision making based on the Z-Wasserstein distance. Expert Systems with Applications. 2023; 214: 14.

48.

Jana

Pal

Liu

. Multiple attribute dynamic decision making method based on some complex aggregation functions in CQROF setting. Computational & Applied Mathematics. 2022; 41(3): 28.

49.

Zhang

Wei

. Location selection of electric vehicles charging stations by using the spherical fuzzy CPT-CoCoSo and D-CRITIC method. Computational & Applied Mathematics. 2023; 42(1): 35.

50.

Pamucar

Biswas

. A novel hybrid decision making framework for comparing market performance of metaverse crypto assets. Decision Making Advances. 2023; 1(1): 49-62.

51.

Zadeh

. Fuzzy Sets. Information and Control. 1965; 338-56.

52.

Atanassov

. Intuitionistic fuzzy sets. Fuzzy Sets and Systems. 1986; 20(1): 87-96.

53.

Smarandache

. A unifying field in logics: Neutrosophic logic. Multiple-Valued Logic. 1999; 8(3).

54.

Wang

Smarandache

Zhang

Sunderraman

. Single valued neutrosophic sets. Multispace Multistruct. 2010; (4): 410-3.

55.

Chakraborty

Saha

. Selection of forklift unit for transport handling using integrated mcdm under neutrosophic environment. Facta Universitatis, Series: Mechanical Engineering. 2022. http://casopisi.junis.ni.ac.rs/index.php/FUMechEng/article/view/10860.

56.

Broumi

Mohanaselvi

Witczak

Talea

Bakali

Smarandache

. Complex fermatean neutrosophic graph and application to decision making. Decision Making: Applications in Management and Engineering. 2023; 6(1): 474-501.

57.

Liu

Yang

. EDAS method for single-valued neutrosophic number multiattribute group decision-making and applications to physical education teaching quality evaluation in colleges and universities. Mathematical Problems in Engineering. 2023; 2023: 5576217.

58.

Chakraborty

Mondal

Alam

Mahata

. Cylindrical neutrosophic single-valued number and its application in networking problem, multi-criterion group decision-making problem and graph theory. CAAI T Intell Technol. 2020; 5(2): 68-77.

59.

Haque

Chakraborty

Mondal

Alam

. A novel logarithmic operational law and aggregation operators for trapezoidal neutrosophic number with MCGDM skill to determine most harmful virus. Applied Intelligence. 2022; 52(4): 4398-417.

60.

Chakraborty

Mondal

Alam

Dey

. Classification of trapezoidal bipolar neutrosophic number, de-bipolarization technique and its execution in cloud service-based MCGDM problem. Complex & Intelligent Systems. 2021; 7(1): 145-62.

61.

Jafar

Saeed

Khan

Alamri

Khalifa

HAW

. Distance and similarity measures using max-min operators of neutrosophic hypersoft sets with application in site selection for solid waste management systems. Ieee Access. 2022; 10: 11220-35.

62.

. Interval-valued intuitionistic fuzzy Heronian mean operators and their application in multi-criteria decision making. African Journal of Business Management. 2012; 6(11): 4158-68.

63.

Zhou

. Intuitionistic Fuzzy Normalized Weighted Bonferroni Mean and Its Application in Multicriteria Decision Making. Journal of Applied Mathematics. 2012.

64.

Beliakov

James

. On extending generalized Bonferroni means to Atanassov orthopairs in decision making contexts. Fuzzy Sets and Systems. 2013; 211: 84-98.

65.

. Intuitionistic fuzzy geometric Heronian mean aggregation operators. Applied Soft Computing. 2013; 13(2): 1235-46.

66.

Liu

Chen

Chu

. Intuitionistic Uncertain Linguistic Weighted Bonferroni OWA Operator and Its Application to Multiple Attribute Decision Making. Cybernetics and Systems. 2014; 45(5): 418-38.

67.

Bonferroni

. Sulle medie multiple di potenze. Bolletino Matematica Italiana. 1950; 5: 267-70.

68.

Yager

. On generalized Bonferroni mean operators for multi-criteria aggregation. International Journal of Approximate Reasoning. 2009; 50(8): 1279-86.

69.

Zhu

Xia

. Hesitant fuzzy geometric Bonferroni means. Information Sciences. 2012; 205: 72-85.

70.

Xia

Zhu

. Generalized intuitionistic fuzzy Bonferroni means. International Journal of Intelligent Systems. 2012; 27(1): 23-47.

71.

Yager

. The power average operator. IEEE Transactions on Systems, Man, and Cybernetics-Part A. 2001; 31(6): 724-31.

72.

Wang

. Using the method of maximizing deviation to make decision for multiindices. Journal of Systems Engineering & Electronics. 1997; 8(3): 21-6.

73.

Peng

Wang

Zhang

Chen

. Simplified neutrosophic sets and their applications in multi-criteria group decision-making problems. International Journal of Systems Science. 2016; 47(10): 2342-58.

74.

Huang

. New distance measure of single-valued neutrosophic sets and its application. International Journal of Intelligent Systems. 2016; 31(10): 1021-32.

75.

Wang

. An Extended Technique for Multiple Attribute Decision Making under Single-Valued Neutrosophic Sets and Applications to Grain Fermentation Process Quality Evaluation. Journal of Intelligent & Fuzzy Systems. 2023. doi: 10.3233/JIFS-231978.

76.

Bolturk

Karasan

. Prioritization of Investment Alternatives for a Hospital by Using Neutrosophic CODAS Method. Journal of Multiple-Valued Logic and Soft Computing. 2019; 33(4-5): 381-96.

77.

Stanujkic

Karabasevic

Popovic

Pamucar

Stevic

Zavadskas

, et al. A Single-Valued Neutrosophic Extension of the EDAS Method. Axioms. 2021; 10(4): 13.

78.

Tversky

. Prospect theory: An analysis of decision under risk. Econometrica. 1979; 47(2): 263-91.

79.

Kahneman

. Advances in prospect theory: cumulative representation of uncertainty. Journal of Risk and Uncertainty. 1992; 5: 297-323.

80.

Bleichrodt

Cillo

Diecidue

. A quantitative measurement of regret theory. Management Science. 2010; 56(1): 161-75.

81.

Pamucar

Deveci

Gokasar

Isik

Zizovic

. Circular economy concepts in urban mobility alternatives using integrated DIBR method and fuzzy Dombi CoCoSo model. Journal of Cleaner Production. 2021; 323: 19.

82.

Pamucar

Simic

Lazarevic

Dobrodolac

Deveci

. Prioritization of sustainable mobility sharing systems using integrated fuzzy DIBR and fuzzy-rough EDAS model. Sustainable Cities and Society. 2022; 82: 30.

83.

Pamucar

Stevic

Sremac

. A New Model for Determining Weight Coefficients of Criteria in MCDM Models: Full Consistency Method (FUCOM). Symmetry-Basel. 2018; 10(9): 393.

84.

Pamucar

Žižović

Biswas

Božanić

. A new logarithm methodology of additive weights (LMAW) for multi-criteria decision-making: Application in logistics. Facta Universitatis, Series: Mechanical Engineering. 2021; 19(3): 361-80.

Smart algorithmic solutions for neutrosophic multiple-attribute decision-making and applications to chair furniture comfort design evaluation

Abstract

Keywords

1. Introduction

2. Preliminaries

5.1 An empirical example

Table 1 The given four attributes

5.3 Comparative analysis

Table 9 Results of different methods

References

Table 1
The given four attributes

Table 9
Results of different methods