Enhancing learning environments with IoT: a novel decision-making approach using probabilistic linguistic T-spherical fuzzy set

Abstract

The Internet of Things is a system of networked devices that can gather, process, and share data through the Internet. The Internet of Things has vast potential to spread widely across various aspects of our lives. The educational process model is undergoing a transformation in which the learning requirements for different students must be fulfilled in various ways. Our study presents a novel multi-attribute group decision-making strategy that examines how the Internet of Things can help to provide the best learning environment while also making the educational process more effective. The probabilistic linguistic T-spherical fuzzy set (PLT-SFS) is a modification of the T-spherical fuzzy set in which the degrees of membership, abstinence, and non-membership are characterized by probabilistic linguistic terms. Then two new aggregation operators, the PLT-SF weighted power average (PLT-SFWPA) operator and the PLT-SF weighted power geometric (PLT-SFWPG) operator, are introduced. After that, an approach to multi-attribute group decision-making based on the analytic hierarchy process is constructed in which the data are aggregated by the PLT-SFWPA operator. To illustrate the validity of the proposed approach, a case study of nine Internet of Things applications for enhancing learning environments is provided.

Appendices

Appendix 1:

Tables 7, 8, 9, 10, 11, and 12 provide a summary of the assessment values assigned by the three DMs to each IoT application. Tables 7, 8, and 9 contain the unnormalized PLT-SF data, and Tables 10, 11, and 12 contain the normalized PLT-SF data.

Table 7 PLT-SF unnormalized decision matrix provided by first DM $\mathcal {D}_1$

Table 8 PLT-SF unnormalized decision matrix provided by second DM $\mathcal {D}_2$

Table 9 PLT-SF unnormalized decision matrix provided by third DM $\mathcal {D}_3$

Table 10 PLT-SF normalized decision matrix provided by first DM $\mathcal {D}_1$

Table 11 PLT-SF normalized decision matrix provided by second DM $\mathcal {D}_2$

Table 12 PLT-SF normalized decision matrix provided by third DM $\mathcal {D}_3$

Appendix 2: Evaluation procedure

The approach employed for the evaluation of nine IoT applications in learning environment is detailed here. The PLT-SF-AHP method in which the data is aggregated by the PLT-SFWPA operator is used for this purpose.

Step 1.:

Construct the PLT-SF evaluation matrix $\hslash ^{\kappa }=[\aleph ^{\kappa }_{{\varrho }{\varsigma }}]_{9\times 4}=\langle \{{\flat }_{\varphi ^{(t)}_{\varrho \varsigma }}(\hat{\mathfrak {d}}_{\varrho \varsigma }^{(t)})\}^\kappa , \{{\flat }_{\phi ^{(r)}_{\varrho \varsigma }}(\tilde{\mathfrak {d}}_{\varrho \varsigma }^{(r)})\}^\kappa , \{{\flat }_{\psi ^{(l)}_{\varrho \varsigma }}(\bar{\mathfrak {d}}_{\varrho \varsigma }^{(l)})\}^\kappa \rangle _{9\times 4}(\varrho =1,2,3,\ldots ,9, \varsigma =1,2,3,4$, and $\kappa =1,2,3)$, which is describing the assessments of three DMs as computed in Tables 7, 8, 9, 10, 11 and 12.

Step 2.:

The purpose of utilizing Eq. (17) is to normalize the decision matrices. Since all attributes possess beneficial qualities, the decision matrices remain unchanged.

Step 3.:

According to Eq. (18), the support degree $\mathbb {S} ({\aleph ^{\kappa }_{\varrho \varsigma }},{\aleph ^{{d}}_{{\varrho }{\varsigma }}})$ can be calculated. For easy to use, $\mathbb {S}({\aleph ^{\kappa }_{\varrho \varsigma }},{\aleph ^{{d}}_{{\varrho }{\varsigma }}})$ can be represented as $\mathbb {S}^{\kappa d}$, shown as follows:

$$\begin{aligned} \mathbb {S}^{12}= & {} \mathbb {S}^{21}= \begin{array}{*{20}c} \mathcal {F}_1 \\ \mathcal {F}_2 \\ \mathcal {F}_3 \\ \mathcal {F}_4\\ \mathcal {F}_5\\ \mathcal {F}_6\\ \mathcal {F}_7\\ \mathcal {F}_8\\ \mathcal {F}_9\\ \end{array} \mathop {\left[ {\begin{array}{*{20}l} 0.9799 \qquad \qquad &{} 0.9517 \qquad \qquad &{} 0.9205 \qquad \qquad &{} 1.0782 \\ 0.9865 \qquad \qquad &{} 1.0761 \qquad \qquad &{} 1.1213 \qquad \qquad &{} 1.1196 \\ 1.0460 \qquad \qquad &{} 1.0927 \qquad \qquad &{} 1.0559 \qquad \qquad &{} 0.9811 \\ 0.8710 \qquad \qquad &{} 0.9547 \qquad \qquad &{} 1.0652 \qquad \qquad &{} 0.8109 \\ 1.0328 \qquad \qquad &{} 0.9611 \qquad \qquad &{} 0.9762 \qquad \qquad &{} 1.0971 \\ 1.0354 \qquad \qquad &{} 0.9515 \qquad \qquad &{} 0.9246 \qquad \qquad &{} 0.9792 \\ 1.0873 \qquad \qquad &{} 1.0117 \qquad \qquad &{} 0.9776 \qquad \qquad &{} 0.9901 \\ 0.9599 \qquad \qquad &{} 0.9892 \qquad \qquad &{} 1.0062 \qquad \qquad &{} 0.9495 \\ 0.9602 \qquad \qquad &{} 0.9254 \qquad \qquad &{} 1.0368 \qquad \qquad &{} 1.0160 \\ \end{array} } \right] }\limits ^{{\begin{array}{*{20}l} &{} \mathcal {N}_1 &{}\qquad \qquad \mathcal {N}_2 &{}\quad \qquad \qquad \mathcal {N}_3 &{}\qquad \qquad \mathcal {N}_4 \\ \end{array} }}\\ \mathbb {S}^{13}= & {} \mathbb {S}^{31} = \begin{array}{*{20}c} \mathcal {F}_1 \\ \mathcal {F}_2 \\ \mathcal {F}_3 \\ \mathcal {F}_4\\ \mathcal {F}_5\\ \mathcal {F}_6\\ \mathcal {F}_7\\ \mathcal {F}_8\\ \mathcal {F}_9\\ \end{array} \mathop {\left[ {\begin{array}{*{20}l} 0.8690 \qquad \qquad &{} 0.8841 \qquad \qquad &{} 0.7282 \qquad \qquad &{} 0.9230 \\ 0.8489 \qquad \qquad &{} 0.9542 \qquad \qquad &{} 0.9197 \qquad \qquad &{} 0.9783 \\ 0.7761 \qquad \qquad &{} 1.1692 \qquad \qquad &{} 0.8669 \qquad \qquad &{} 0.9164 \\ 0.8785 \qquad \qquad &{} 0.8550 \qquad \qquad &{} 0.9600 \qquad \qquad &{} 0.8231 \\ 1.0032 \qquad \qquad &{} 0.9034 \qquad \qquad &{} 0.8212 \qquad \qquad &{} 0.7761 \\ 0.9604 \qquad \qquad &{} 0.9479 \qquad \qquad &{} 0.8539 \qquad \qquad &{} 0.8285 \\ 0.9881 \qquad \qquad &{} 0.8519 \qquad \qquad &{} 0.8090 \qquad \qquad &{} 0.6286 \\ 0.9343 \qquad \qquad &{} 0.8526 \qquad \qquad &{} 0.8401 \qquad \qquad &{} 0.8628 \\ 0.8300 \qquad \qquad &{} 0.8779 \qquad \qquad &{} 1.0922 \qquad \qquad &{} 0.9157 \\ \end{array} } \right] }\limits ^{{\begin{array}{*{20}l} \mathcal {N}_1 &{}\qquad \qquad \mathcal {N}_2 &{} \quad \qquad \qquad \mathcal {N}_3 &{} \qquad \qquad \mathcal {N}_4 \\ \end{array} }} \end{aligned}$$

  

$$\begin{aligned} ~ \mathbb {S}^{23}= \mathbb {S}^{32}= \begin{array}{*{20}c} \mathcal {F}_1 \\ \mathcal {F}_2 \\ \mathcal {F}_3 \\ \mathcal {F}_4 \\ \mathcal {F}_5\\ \mathcal {F}_6\\ \mathcal {F}_7\\ \mathcal {F}_8\\ \mathcal {F}_9\\ \end{array} \mathop {\left[ {\begin{array}{*{20}l} 0.7704 \qquad \qquad &{} 0.8770 \qquad \qquad &{} 0.8314 \qquad \qquad &{} 0.8168\\ 0.8437 \qquad \qquad &{} 0.8981 \qquad \qquad &{} 0.7536 \qquad \qquad &{} 0.9201\\ 0.8632 \qquad \qquad &{} 1.0696 \qquad \qquad &{} 0.8748 \qquad \qquad &{} 0.9071\\ 1.0261 \qquad \qquad &{} 0.9509 \qquad \qquad &{} 0.8364 \qquad \qquad &{} 0.9395\\ 0.8770 \qquad \qquad &{} 0.9424 \qquad \qquad &{} 0.7950 \qquad \qquad &{} 0.6888\\ 0.9419 \qquad \qquad &{} 1.0805 \qquad \qquad &{} 0.9260 \qquad \qquad &{} 0.8371\\ 0.9069 \qquad \qquad &{} 0.9520 \qquad \qquad &{} 0.7702 \qquad \qquad &{} 0.7706\\ 0.9049 \qquad \qquad &{} 0.8350 \qquad \qquad &{} 0.7844 \qquad \qquad &{} 0.9113\\ 0.8004 \qquad \qquad &{} 0.9048 \qquad \qquad &{} 1.0284 \qquad \qquad &{} 0.7140\\ \end{array} } \right] }\limits ^{{\begin{array}{*{20}l} &{} \mathcal {N}_1 &{} \qquad \qquad \mathcal {N}_2 &{}\quad \qquad \qquad \mathcal {N}_3 &{}\qquad \qquad \mathcal {N}_4 \\ \end{array} }} \end{aligned}$$

Step 4.:

According to Eq. (19), the collective support matrices $\mathbb {T}({\aleph ^{\kappa }_{{\varrho }{\varsigma }}})$ of the PLT-SFN can be calculated. For easy to use, $\mathbb {T}({\aleph ^{\kappa }_{{\varrho }{\varsigma }}})$ can be represented by a matrix $\mathbb {T}({\aleph ^{\kappa }})$, shown as follows:

$$\begin{aligned} \mathbb {T}^{1}= & {} \begin{array}{*{20}c} \mathcal {F}_1\\ \mathcal {F}_2\\ \mathcal {F}_3\\ \mathcal {F}_4\\ \mathcal {F}_5\\ \mathcal {F}_6\\ \mathcal {F}_7\\ \mathcal {F}_8\\ \mathcal {F}_9\\ \end{array} \mathop {\left[ {\begin{array}{*{20}l} 1.8490 \qquad \qquad &{} 1.8357 \qquad \qquad &{} 1.6488 \qquad \qquad &{} 2.0012\\ 1.8354 \qquad \qquad &{} 2.0303 \qquad \qquad &{} 2.0410 \qquad \qquad &{} 2.0979\\ 1.8221 \qquad \qquad &{} 2.2620 \qquad \qquad &{} 1.9228 \qquad \qquad &{} 1.8975\\ 1.7496 \qquad \qquad &{} 1.8098 \qquad \qquad &{} 2.0252 \qquad \qquad &{} 1.6340\\ 2.0360 \qquad \qquad &{} 1.8645 \qquad \qquad &{} 1.7974 \qquad \qquad &{} 1.8733\\ 1.9958 \qquad \qquad &{} 1.8993 \qquad \qquad &{} 1.7786 \qquad \qquad &{} 1.8077\\ 2.0754 \qquad \qquad &{} 1.8636 \qquad \qquad &{} 1.7866 \qquad \qquad &{} 1.6187\\ 1.8942 \qquad \qquad &{} 1.8418 \qquad \qquad &{} 1.8463 \qquad \qquad &{} 1.8123\\ 1.7902 \qquad \qquad &{} 1.8033 \qquad \qquad &{} 2.1290 \qquad \qquad &{} 1.9317\\ \end{array} } \right] }\limits ^{{\begin{array}{*{20}l} &{} \mathcal {N}_1 &{}\qquad \qquad \mathcal {N}_2 &{}\quad \qquad \qquad \mathcal {N}_3 &{} \qquad \qquad \mathcal {N}_4 \\ \end{array} }}\\ \mathbb {T}^{2}= & {} \begin{array}{*{20}c} \mathcal {F}_1\\ \mathcal {F}_2\\ \mathcal {F}_3\\ \mathcal {F}_4\\ \mathcal {F}_5\\ \mathcal {F}_6\\ \mathcal {F}_7\\ \mathcal {F}_8\\ \mathcal {F}_9\\ \end{array} \mathop {\left[ {\begin{array}{*{20}l} 1.7503 \qquad \qquad &{} 1.8287 \qquad \qquad &{} 1.7520 \qquad \qquad &{} 1.8950 \\ 1.8302 \qquad \qquad &{} 1.9743 \qquad \qquad &{} 1.8749 \qquad \qquad &{} 2.0398 \\ 1.9092 \qquad \qquad &{} 2.1623 \qquad \qquad &{} 1.9308 \qquad \qquad &{} 1.8882 \\ 1.8971 \qquad \qquad &{} 1.9056 \qquad \qquad &{} 1.9016 \qquad \qquad &{} 1.7503 \\ 1.9098 \qquad \qquad &{} 1.9035 \qquad \qquad &{} 1.7712 \qquad \qquad &{} 1.7859 \\ 1.9772 \qquad \qquad &{} 2.0320 \qquad \qquad &{} 1.8507 \qquad \qquad &{} 1.8163 \\ 1.9941 \qquad \qquad &{} 1.9637 \qquad \qquad &{} 1.7479 \qquad \qquad &{} 1.7607 \\ 1.8648 \qquad \qquad &{} 1.8242 \qquad \qquad &{} 1.7906 \qquad \qquad &{} 1.8608 \\ 1.7606 \qquad \qquad &{} 1.8302 \qquad \qquad &{} 2.0652 \qquad \qquad &{} 1.7300 \\ \end{array} } \right] }\limits ^{{\begin{array}{*{20}l} &{} \mathcal {N}_1 &{}\qquad \qquad \mathcal {N}_2 &{}\qquad \qquad \quad \mathcal {N}_3 &{}\qquad \qquad \mathcal {N}_4 \\ \end{array} }}\\ \mathbb {T}^{3}= & {} \begin{array}{*{20}c} \mathcal {F}_1\\ \mathcal {F}_2\\ \mathcal {F}_3\\ \mathcal {F}_4\\ \mathcal {F}_5\\ \mathcal {F}_6\\ \mathcal {F}_7 \\ \mathcal {F}_8\\ \mathcal {F}_9\\ \end{array} \mathop {\left[ {\begin{array}{*{20}l} 1.6394 \qquad \qquad &{} 1.7611 \qquad \qquad &{} 1.5596 \qquad \qquad &{} 1.7397\\ 1.6926 \qquad \qquad &{} 1.8523 \qquad \qquad &{} 1.6733 \qquad \qquad &{} 1.8985\\ 1.6393 \qquad \qquad &{} 2.2388 \qquad \qquad &{} 1.7417 \qquad \qquad &{} 1.8235\\ 1.9047 \qquad \qquad &{} 1.8059 \qquad \qquad &{} 1.7965 \qquad \qquad &{} 1.7625\\ 1.8802 \qquad \qquad &{} 1.8457 \qquad \qquad &{} 1.6162 \qquad \qquad &{} 1.4649\\ 1.9022 \qquad \qquad &{} 2.0284 \qquad \qquad &{} 1.7799 \qquad \qquad &{} 1.6656\\ 1.8949 \qquad \qquad &{} 1.8039 \qquad \qquad &{} 1.5792 \qquad \qquad &{} 1.3992\\ 1.8392 \qquad \qquad &{} 1.6876 \qquad \qquad &{} 1.6245 \qquad \qquad &{} 1.7741\\ 1.6304 \qquad \qquad &{} 1.7827 \qquad \qquad &{} 2.1206 \qquad \qquad &{} 1.6296\\ \end{array} } \right] }\limits ^{{\begin{array}{*{20}l} &{} \mathcal {N}_1 &{}\qquad \qquad \mathcal {N}_2 &{}\qquad \qquad \quad \mathcal {N}_3 &{}\qquad \qquad \mathcal {N}_4 \\ \end{array} }} \end{aligned}$$

Step 5.:

According to Eq. (20), the power weight matrix of DM $\mathcal {D}_{\kappa }$ associated with the PLT-SFN can be calculated. For easy to use, $\mathfrak {\zeta }({\aleph ^{\kappa }_{{\varrho }{\varsigma }}})$ can be represented by a matrix $\mathfrak {\zeta }({\aleph ^{\kappa }})$, shown as follows:

$$\begin{aligned} \zeta ^{1}= & {} \begin{array}{*{20}c} \mathcal {F}_1\\ \mathcal {F}_2\\ \mathcal {F}_3\\ \mathcal {F}_4\\ \mathcal {F}_5 \\ \mathcal {F}_6\\ \mathcal {F}_7\\ \mathcal {F}_8 \\ \mathcal {F}_9\\ \end{array} \mathop {\left[ {\begin{array}{*{20}l} 0.3732 \qquad \qquad &{} 0.3638 \qquad \qquad &{} 0.3596 \qquad \qquad &{} 0.3750 \\ 0.3664 \qquad \qquad &{} 0.3695 \qquad \qquad &{} 0.3816 \qquad \qquad &{} 0.3702 \\ 0.3641 \qquad \qquad &{} 0.3650 \qquad \qquad &{} 0.3672 \qquad \qquad &{} 0.3637 \\ 0.3483 \qquad \qquad &{} 0.3566 \qquad \qquad &{} 0.3743 \qquad \qquad &{} 0.3501 \\ 0.3715 \qquad \qquad &{} 0.3596 \qquad \qquad &{} 0.3690 \qquad \qquad &{} 0.3810 \\ 0.3648 \qquad \qquad &{} 0.3504 \qquad \qquad &{} 0.3574 \qquad \qquad &{} 0.3658 \\ 0.3704 \qquad \qquad &{} 0.3587 \qquad \qquad &{} 0.3707 \qquad \qquad &{} 0.3634 \\ 0.3639 \qquad \qquad &{} 0.3674 \qquad \qquad &{} 0.3718 \qquad \qquad &{} 0.3599 \\ 0.3683 \qquad \qquad &{} 0.3602 \qquad \qquad &{} 0.3633 \qquad \qquad &{} 0.3813 \\ \end{array} } \right] }\limits ^{{\begin{array}{*{20}l} &{} \mathcal {N}_1 &{}\qquad \qquad \mathcal {N}_2 &{}\quad \qquad \qquad \mathcal {N}_3 &{}\qquad \qquad \mathcal {N}_4 \\ \end{array} }}\\ \zeta ^{2}= & {} \begin{array}{*{20}c} \mathcal {F}_1\\ \mathcal {F}_2\\ \mathcal {F}_3\\ \mathcal {F}_4\\ \mathcal {F}_5\\ \mathcal {F}_6\\ \mathcal {F}_7\\ \mathcal {F}_8\\ \mathcal {F}_9\\ \end{array} \mathop {\left[ {\begin{array}{*{20}l} 0.3310 \qquad \qquad &{} 0.3334 \qquad \qquad &{} 0.3432 \qquad \qquad &{} 0.3323 \\ 0.3360 \qquad \qquad &{} 0.3332 \qquad \qquad &{} 0.3315 \qquad \qquad &{} 0.3337 \\ 0.3448 \qquad \qquad &{} 0.3251 \qquad \qquad &{} 0.3383 \qquad \qquad &{} 0.3331 \\ 0.3371 \qquad \qquad &{} 0.3388 \qquad \qquad &{} 0.3298 \qquad \qquad &{} 0.3359 \\ 0.3271 \qquad \qquad &{} 0.3349 \qquad \qquad &{} 0.3358 \qquad \qquad &{} 0.3394 \\ 0.3330 \qquad \qquad &{} 0.3366 \qquad \qquad &{} 0.3368 \qquad \qquad &{} 0.3371 \\ 0.3313 \qquad \qquad &{} 0.3410 \qquad \qquad &{} 0.3358 \qquad \qquad &{} 0.3519 \\ 0.3309 \qquad \qquad &{} 0.3354 \qquad \qquad &{} 0.3349 \qquad \qquad &{} 0.3364 \\ 0.3348 \qquad \qquad &{} 0.3341 \qquad \qquad &{} 0.3269 \qquad \qquad &{} 0.3262 \\ \end{array} } \right] }\limits ^{{\begin{array}{*{20}l} &{} \mathcal {N}_1 &{}\qquad \qquad \mathcal {N}_2 &{}\qquad \qquad \quad \mathcal {N}_3 &{}\qquad \qquad \mathcal {N}_4 \\ \end{array} }}\\ \zeta ^{3}= & {} \begin{array}{*{20}c} \mathcal {F}_1\\ \mathcal {F}_2\\ \mathcal {F}_3\\ \mathcal {F}_4\\ \mathcal {F}_5\\ \mathcal {F}_6\\ \mathcal {F}_7\\ \mathcal {F}_8\\ \mathcal {F}_9\\ \end{array} \mathop {\left[ {\begin{array}{*{20}l} 0.2957 \qquad \qquad &{} 0.3029 \qquad \qquad &{} 0.2972 \qquad \qquad &{} 0.2927 \\ 0.2976 \qquad \qquad &{} 0.2974 \qquad \qquad &{} 0.2869 \qquad \qquad &{} 0.2962 \\ 0.2912 \qquad \qquad &{} 0.3099 \qquad \qquad &{} 0.2946 \qquad \qquad &{} 0.3031 \\ 0.3146 \qquad \qquad &{} 0.3046 \qquad \qquad &{} 0.2959 \qquad \qquad &{} 0.3140 \\ 0.3014 \qquad \qquad &{} 0.3055 \qquad \qquad &{} 0.2951 \qquad \qquad &{} 0.2795 \\ 0.3022 \qquad \qquad &{} 0.3130 \qquad \qquad &{} 0.3058 \qquad \qquad &{} 0.2970 \\ 0.2982 \qquad \qquad &{} 0.3003 \qquad \qquad &{} 0.2934 \qquad \qquad &{} 0.2847 \\ 0.3053 \qquad \qquad &{} 0.2972 \qquad \qquad &{} 0.2932 \qquad \qquad &{} 0.3037 \\ 0.2969 \qquad \qquad &{} 0.3058 \qquad \qquad &{} 0.3098 \qquad \qquad &{} 0.2925 \\ \end{array} } \right] }\limits ^{{\begin{array}{*{20}l} &{} \mathcal {N}_1 &{}\qquad \qquad \mathcal {N}_2 &{}\quad \qquad \qquad \mathcal {N}_3 &{}\qquad \qquad \mathcal {N}_4 \\ \end{array} }} \end{aligned}$$

Step 6.:

To combine the values from overall ${\aleph }_{{\varrho }{\varsigma }}^{\kappa }$ to ${\aleph }_{\varrho \varsigma }$, the PLT-SFWPA operator is employed represented by Eq. (13). The resulting comprehensive PLT-SFNs matrix, denoted as ${\hslash }=[{\aleph }_{{\varrho }{\varsigma }}]_{\mathfrak {a}\times \mathfrak {b}}$ is illustrated below:

$$\begin{aligned}{} & {} \begin{array}{ll} \text {Alternatives}=\mathcal {N}_{1} \\ {\mathcal {F}_{1}}=\langle \{\flat _{1.5082}(0),\flat _{2.0909}(0.0560),\flat _{4.0000}(0.0240)\},\{\flat _{-1.5746}(0),\flat _{-0.5108}(0.0080),\\ ~~~~~~~~~\flat _{ 2.9903}( 0.3240)\},\{\flat _{-4.0000}(0),\flat _{-1.8896}(0.0160), \flat _{-0.2089}(0.2160)\}\rangle \\ {\mathcal {F}_{2}}=\langle \{\flat _{0.6438}(0),\flat _{1.6732}(0.0400),\flat _{4.0000}(0.0311)\},\{\flat _{-2.0000}(0.0100),\flat _{ 0.8817}(0.1800),\\ ~~~~~~~~~\flat _{ 1.9262}(0.0100)\}, \{\flat _{-1.7235}(0.0080),\flat _{-0.5722}( 0.0080),\flat _{ 1.4400}( 0.2160)\}\rangle \\ {\mathcal {F}_{3}}=\langle \{\flat _{0.0398}(0),\flat _{0.6155}(0.0720),\flat _{2.6938}(0.0640)\},\{\flat _{-2.7300}(0),\flat _{-1.4259}(0.0600),\\ ~~~~~~~~~ \flat _{ 1.1711}( 0.0600)\}, \{\flat _{-1.7494}(0.0133),\flat _{-0.7379}(0.0355),\flat _{ 1.3431}(0.0710)\}\rangle \\ {\mathcal {F}_{4}}=\langle \{\flat _{-0.1052}(0),\flat _{0.0888}(0.0040),\flat _{2.1079}(0.0360)\},\{\flat _{-4.0000}(0.0320),\flat _{0.2090}(0.0640),\\ ~~~~~~~~~\flat _{ 2.3200}(0.0160)\}, \{\flat _{-4.0000}(0.0240),\flat _{-1.9286}(0.0320),\flat _{-0.1522}(0.0320)\}\rangle \\ {\mathcal {F}_{5}}=\langle \{\flat _{-3.2193}(0.0640),\flat _{-0.6380}(0.0320),\flat _{2.2857}(0.0160)\}, \{\flat _{-1.3639}(0),\flat _{-0.5897}(0.0240),\\ ~~~~~~~~~\flat _{ 2.6822}(0.0640)\}, \{\flat _{-2.6075}(0),\flat _{-1.8247}(0.1280),\flat _{-0.2239}(0.0320)\}\rangle \\ {\mathcal {F}_{6}}=\langle \{\flat _{-1.7718}(0.0160),\flat _{-0.1224}(0.0640),\flat _{1.0407}(0.0320)\}, \{\flat _{-4.0000}(0.0320),\flat _{-0.9805}( 0.0320),\\ ~~~~~~~~~\flat _{ 2.3232}(0.0320)\}, \{\flat _{-0.6984}(0.0160),\flat _{0.3086}(0.0640),\flat _{2.3232}(0.0320)\}\rangle \\ {\mathcal {F}_{7}}=\langle \{\flat _{0.0593}(0),\flat _{1.1056}(0.0120),\flat _{4.0000}(0.1680)\}, \{\flat _{-4.0000}(0.2002),\\ ~~~~~~~~~\flat _{-2.1099}(0.0089),\flat _{ 0.9214}(0.0089)\}, \{\flat _{-4.0000}(0.0640),\flat _{-1.7124}(0.0160),\flat _{-0.7000}(0.0320)\}\rangle \\ {\mathcal {F}_{8}}=\langle \{\flat _{-0.7690}(0.0160),\flat _{0.3935}(0.0160),\flat _{2.7858}(0.0720)\}, \{\flat _{-4.0000}(0.0667),\flat _{0.3383}( 0.0266),\\ ~~~~~~~~~\flat _{1.6752}(0.0089)\}, \{\flat _{-4.0000}(0.0640),\flat _{-0.8917}(0.0160),\flat _{ 1.6512}(0.0320)\}\rangle \\ {\mathcal {F}_{9}}=\langle \{\flat _{-1.4842}(0.0089),\flat _{0.1289}(0.0089),\flat _{2.3396}(0.1601)\}, \{\flat _{-4.0000}( 0.2002),\flat _{-1.0714}(0.0089),\\ ~~~~~~~~~\flat _{ 2.0000}(0.00890)\}, \{\flat _{-4.0000}(0),\flat _{-1.7876}(0.0400),\flat _{ 0.8256}(0.0400)\}\rangle \end{array} \\ & {} \begin{array}{ll} \text {Alternatives}=\mathcal {N}_{2} \\ {\mathcal {F}_{1}}=\langle \{\flat _{0.6119}(0),\flat _{1.1101}(0.0932),\flat _{2.2361}(0.0400)\}, \{\flat _{-4.0000}(0.0240),\flat _{-0.7269}(0.0080),\\ ~~~~~~~~~ \flat _{ 1.6776}(0.0720)\}, \{\flat _{-4.0000}(0),\flat _{-4.0000}(0.0240),\flat _{ 1.4155}(0.1120)\}\rangle \\ {\mathcal {F}_{2}}=\langle \{\flat _{-1.7724}(0.0640),\flat _{0.9498}(0.0320),\flat _{2.7334}(0.0160)\}, \{\flat _{-4.0000}(0.0640),\flat _{-0.7320}( 0.0320),\\ ~~~~~~~~~\flat _{ 0.5154}(0.0160)\}, \{\flat _{-1.6768}(0),\flat _{-0.5078}(0.0160),\flat _{ 1.9828}(0.2160)\}\rangle \\ {\mathcal {F}_{3}}=\langle \{\flat _{ 0.7413}(0.0080),\flat _{1.9003}(0.2160),\flat _{4.0000}(0.0080)\}, \{\flat _{-4.0000}(0),\flat _{-4.0000}(0.0222),\\ ~~~~~~~~~\flat _{ 0.7476}(0.0197)\}, \{\flat _{-3.0000}(0.0640),\flat _{-2.0000}(0.0640),\flat _{ 0.1495}(0.0080)\}\rangle \\ {\mathcal {F}_{4}}=\langle \{\flat _{-3.2571}(0.0080),\flat _{0.2052}(0.0640),\flat _{1.1900}(0.0640)\}, \{\flat _{-4.0000}(0.0710),\flat _{ 0.1673}( 0.0355),\\ ~~~~~~~~~\flat _{ 1.4706}(0.0133)\}, \{\flat _{-4.0000}(0),\flat _{-2.1287}(0.0480),\flat _{ 0.7577}(0.0960)\}\rangle \\ {\mathcal {F}_{5}}=\langle \{\flat _{-0.7443}(0.0640),\flat _{0.2047}(0.0080),\flat _{2.4756}(0.0640)\}, \{\flat _{-1.7954}(0),\flat _{ 1.0000}(0.2001), \\ ~~~~~~~~~ \flat _{ 2.0000}(0.0222)\}, \{\flat _{-2.0000}(0.0240),\flat _{ 0.2775}(0.0080),\flat _{ 1.8999}(0.0720)\}\rangle \\ {\mathcal {F}_{6}}=\langle \{\flat _{-1.6426}(0.0240),\flat _{0.7506}(0.0080),\flat _{2.4240}(0.0720)\}, \{\flat _{-4.0000}(0),\flat _{-1.4502}(0.0240),\\ ~~~~~~~~~\flat _{ 1.2461}(0.0160)\}, \{\flat _{-4.0000}(0.0640),\flat _{-1.0773}(0.0080),\flat _{ 0.5423}(0.0640)\}\rangle \\ {\mathcal {F}_{7}}=\langle \{\flat _{-1.2542}(0),\flat _{0.2501}(0.0100),\flat _{4.0000}(0.0900)\}, \{\flat _{-4.0000}(0.0050),\flat _{-1.6869}(0.0750),\\ ~~~~~~~~~\flat _{ 1.1766}(0.0300)\}, \{\flat _{-4.0000}(0.0160),\flat _{-1.8431}(0.0320),\flat _{-0.4949}(0.0480)\}\rangle \\ {\mathcal {F}_{8}}=\langle \{\flat _{-0.6699}(0),\flat _{0.9053}(0.0300),\flat _{2.1479}(0.0350)\}, \{\flat _{-1.7866}(0.0080),\flat _{-0.6444}(0.0720),\\ ~~~~~~~~~\flat _{ 1.4419}(0.0160)\}, \{\flat _{-1.9247}(0.0160),\flat _{-0.8298}(0.0320),\flat _{ 1.9852}(0.0480)\}\rangle \\ {\mathcal {F}_{9}}=\langle \{\flat _{-3.1056}(0.0320),\flat _{0.1128}(0.0480),\flat _{2.4477}(0.0640)\}, \{\flat _{-4.0000}(0),\flat _{-2.7164}( 0.0266),\\ ~~~~~~~~~\flat _{-0.8596}(0.0400)\}, \{\flat _{-2.7639}(0),\flat _{-1.2180}(0.0320),\flat _{ 1.9746}(0.0480)\}\rangle \end{array}\end{aligned}$$

$$\begin{aligned}{} & {} \\{} & {} \begin{array}{lcccccccccccccccccc} \text {Alternatives}=\mathcal {N}_{3} \\ {\mathcal {F}_{1}}=\langle \{\flat _{-1.6732}(0),\flat _{-1.0000}(0.0320),\flat _{4.0000}(0.1280)\},\\ ~~~~~~~~~\{\flat _{-4.0000}(0.0720),\flat _{-1.3406}( 0.0080),\flat _{0.2206}(0.0240)\},\\ ~~~~~~~~~\{\flat _{-4.0000}(0.0080),\flat _{-0.3931}(0.0080),\flat _{1.1794}(0.2160)\}\rangle \\ {\mathcal {F}_{2}}=\langle \{\flat _{-0.0299}(0.0640),\flat _{1.1902}(0.0640),\flat _{2.2991}(0.0080)\},\\ ~~~~~~~~~\{\flat _{-4.0000}(0),\flat _{-4.0000}( 0.0640),\flat _{1.6851}(0.0240)\},\\ ~~~~~~~~~\{\flat _{-2.1249}(0),\flat _{-1.2457}(0.2160),\flat _{0.6073}(0.0160 )\}\rangle \\ {\mathcal {F}_{3}}=\langle \{\flat _{-1.6610}(0.0640),\flat _{-0.0000}(0.0320),\flat _{2.6332}(0.0160)\},\\ ~~~~~~~~~\{\flat _{-4.0000}(0.0080),\flat _{ 0.0601}( 0.2160),\flat _{1.9906}(0.0080)\},\\ ~~~~~~~~~\{\flat _{-4.0000}(0),\flat _{-2.7358}(0.0640),\flat _{0.8399}(0.0240 )\}\rangle \\ {\mathcal {F}_{4}}=\langle \{\flat _{-4.0000}(0.0111),\flat _{-2.3959}(0.0333),\flat _{2.7661}(0.0888)\},\\ ~~~~~~~~~\{\flat _{-4.0000}(0.0133),\flat _{-0.4083}( 0.0133),\flat _{1.1392}(0.1199)\},\\ ~~~~~~~~~\{\flat _{-2.4570}(0.0240),\flat _{-0.9494}(0.0080),\flat _{1.9318}(0.0720 )\}\rangle \\ {\mathcal {F}_{5}}=\langle \{\flat _{-1.2252}(0.1199),\flat _{-0.6382}(0.0133),\flat _{1.1037}(0.0133)\},\\ ~~~~~~~~~\{\flat _{-1.9257}(0.0089),\flat _{-0.8906}( 0.2002),\flat _{2.6887}(0.0089)\},\\ ~~~~~~~~~\{\flat _{-4.0000}(0),\flat _{-4.0000}(0.0320),\flat _{1.4512}(0.0480 )\}\rangle \\ {\mathcal {F}_{6}}=\langle \{\flat _{-1.0089}(0.0100),\flat _{ 0.3941}(0.0100),\flat _{2.7854}(0.1800)\},\\ ~~~~~~~~~\{\flat _{-1.0000}(0),\flat _{-0.3694}( 0.0280),\flat _{1.6747}(0.0120)\},\\ ~~~~~~~~~\{\flat _{-4.0000}(0.0640),\flat _{ 0}(0.0080),\flat _{1.6389}(0.0640 )\}\rangle \\ {\mathcal {F}_{7}}=\langle \{\flat _{-2.1661}(0.0320),\flat _{0.7410}(0.0320),\flat _{2.7951}(0.0320)\},\\ ~~~~~~~~~\{\flat _{-1.7744}(0),\flat _{0.3040}( 0.2160),\flat _{2.3529}(0.0160)\},\\ ~~~~~~~~~\{\flat _{-4.0000}(0.0160),\flat _{-1.1296}(0.0160),\flat _{0.8467}(0.0960 )\}\rangle \\ {\mathcal {F}_{8}}=\langle \{\flat _{-0.1482}(0.0320),\flat _{ 1.0601}(0.0320),\flat _{2.0928}(0.0320)\},\\ ~~~~~~~~~\{\flat _{-4.0000}( 0.0266),\flat _{-0.1269}(0.0799),\flat _{0.8917}(0.0133)\},\\ ~~~~~~~~~\{\flat _{-4.0000}(0.0040),\flat _{-0.8392}(0.0160),\flat _{1.7380}(0.1680 )\}\rangle \\ {\mathcal {F}_{9}}=\langle \{\flat _{0.8708}(0),\flat _{ 1.8619}(0.0960),\flat _{4.0000}(0.0480)\},\\ ~~~~~~~~~\{\flat _{-1.9348}(0),\flat _{ 0.8939}( 0.1399),\flat _{2.3101}(0.0200)\},\\ ~~~~~~~~~\{\flat _{-4.0000}(0.0160),\flat _{-1.8655}(0.0160),\flat _{0.2690}(0.0960 )\}\rangle \end{array} \\{} & {} \begin{array}{lcccccccccccccccccc} \text {Alternatives}=\mathcal {N}_{4} \\ {\mathcal {F}_{1}}=\langle \{\flat _{-0.0374}(0),\flat _{1.4218}(0.1120),\flat _{2.4457}(0.0480)\},\\ ~~~~~~~~~\{\flat _{-2.7750}(0),\flat _{-2.0000}(0.1598),\flat _{ 2.6911}( 0.0178)\},\\ ~~~~~~~~~\{\flat _{-4.0000}(0),\flat _{-0.8785}(0.0040),\flat _{0.3499}(0.1280)\}\rangle \\ {\mathcal {F}_{2}}=\langle \{\flat _{-0.6861}(0.0640),\flat _{ 1.1931}(0.0640),\flat _{2.6393}(0.0080)\},\\ ~~~~~~~~~\{\flat _{-4.0000}(0.0044),\flat _{ 0.6802}( 0.0355),\flat _{ 1.9847}(0.1066)\},\\ ~~~~~~~~~\{\flat _{-4.0000}(0),\flat _{-2.0024}(0.0320),\flat _{-0.4568}(0.0720)\}\rangle \\ {\mathcal {F}_{3}}=\langle \{\flat _{-1.6542}(0.0160),\flat _{-0.5266}(0.0240),\flat _{2.7875}(0.1120)\},\\ ~~~~~~~~~\{\flat _{-4.0000}(0.1855),\flat _{-2.5087}( 0.0099),\flat _{-1.0175}(0.0099)\},\\ ~~~~~~~~~\{\flat _{-3.0000}(0.0640),\flat _{-2.0000}(0.0640),\flat _{ 2.3212}(0.0080)\}\rangle \\ {\mathcal {F}_{4}}=\langle \{\flat _{-2.0000}(0.0888),\flat _{ 0.0926}(0.0333),\flat _{2.2962}(0.0111)\},\\ ~~~~~~~~~\{\flat _{-4.0000}(0),\flat _{-0.8692}(0.0080),\flat _{ 1.6251}(0.1280)\},\\ ~~~~~~~~~\{\flat _{-1.6949}(0.0080),\flat _{-0.6821}(0.0640),\flat _{ 2.2981}(0.0640)\}\rangle \\ {\mathcal {F}_{5}}=\langle \{\flat _{-2.1461}(0),\flat _{-0.3052}(0.0200),\flat _{2.7082}(0.0800)\},\\ ~~~~~~~~~\{\flat _{-2.1536}(0),\flat _{-0.1299}(0.2002),\flat _{ 1.7019}( 0.0178)\},\\ ~~~~~~~~~\{\flat _{-4.0000}(0.0080),\flat _{0.3346}(0.0080),\flat _{2.0081}(0.2160)\}\rangle \\ {\mathcal {F}_{6}}=\langle \{\flat _{-2.2149}(0.0100),\flat _{-1.0000}(0.0100),\flat _{1.7740}(0.1800)\},\\ ~~~~~~~~~\{\flat _{-4.0000}(0.2160),\flat _{-0.2237}( 0.0080),\flat _{ 2.3481}(0.0080)\},\\ ~~~~~~~~~\{\flat _{-4.0000}(0),\flat _{-0.9000}(0.0240),\flat _{0.8250}(0.0640)\}\rangle \\ {\mathcal {F}_{7}}=\langle \{\flat _{-0.2680}(0.0640),\flat _{0.9206}(0.0160),\flat _{2.5993}(0.0320)\},\\ ~~~~~~~~~\{\flat _{-1.3782}(0),\flat _{ 1.0136}(0.0360),\flat _{ 2.0247}(0.0840)\},\\ ~~~~~~~~~\{\flat _{-4.0000}(0.0040),\flat _{0.1157}(0.0080),\flat _{1.4996}(0.2520)\}\rangle \\ {\mathcal {F}_{8}}=\langle \{\flat _{-1.2538}(0.0089),\flat _{0.2742}(0.1601),\flat _{2.6781}(0.0089)\},\\ ~~~~~~~~~\{\flat _{-4.0000}(0.0080),\flat _{-1.2186}( 0.2160),\flat _{ 1.1199}(0.0080)\},\\ ~~~~~~~~~\{\flat _{-2.0765}(0),\flat _{-1.1707}(0.0640),\flat _{1.6438}(0.0480)\}\rangle \\ {\mathcal {F}_{9}}=\langle \{\flat _{-0.2442}(0),\flat _{0.2069}(0.0080),\flat _{4.0000}(0.1920)\},\\ ~~~~~~~~~\{\flat _{-1.7482}(0),\flat _{-1.3717}(0.0840),\flat _{1.2854}( 0.0480)\},\\ ~~~~~~~~~\{\flat _{-4.0000}(0),\flat _{-0.7366}(0.0160),\flat _{0.9837}(0.0960)\}\rangle \end{array} \end{aligned}$$

Step 7. Construct a pairwise comparison matrix for each attribute by comparing each attribute against every other attribute in terms of their relative importance as:

$$\begin{aligned} \mathbb {R}= \begin{array}{*{20}c} \mathcal {N}_1\\ \mathcal {N}_2 \\ \mathcal {N}_3\\ \mathcal {N}_4\\ \end{array} \mathop {\left[ {\begin{array}{*{20}l} 1.0\qquad \qquad &{} 0.6\qquad \qquad &{} 0.4 \qquad \qquad &{}0.8\\ 0.4\qquad \qquad &{} 1.0\qquad \qquad &{} 0.7 \qquad \qquad &{}0.5\\ 0.7\qquad \qquad &{} 0.3\qquad \qquad &{} 1.0 \qquad \qquad &{}0.6\\ 0.2\qquad \qquad &{} 0.8\qquad \qquad &{} 0.5 \qquad \qquad &{}1.0\\ \end{array} } \right] }\limits ^{{\begin{array}{*{20}l} &{} \mathcal {N}_1 &{} \qquad \mathcal {N}_2 &{} \qquad \quad \mathcal {N}_3 &{} \qquad \mathcal {N}_4 \\ \end{array} }} \end{aligned}$$

Step 8. Normalize the pairwise comparison matrix as:

$$\begin{aligned} \mathbb {N}= \begin{array}{*{20}c} \mathcal {N}_1\\ \mathcal {N}_2\\ \mathcal {N}_3\\ \mathcal {N}_4\\ \end{array} \mathop {\left[ {\begin{array}{*{20}l} 0.3571 \qquad \qquad &{} 0.2143 \qquad \qquad &{} 0.1429 \qquad \qquad &{} 0.2857\\ 0.1538 \qquad \qquad &{} 0.3846 \qquad \qquad &{} 0.2692 \qquad \qquad &{} 0.1923\\ 0.2692 \qquad \qquad &{} 0.1154 \qquad \qquad &{} 0.3846 \qquad \qquad &{} 0.2308\\ 0.0800 \qquad \qquad &{} 0.3200 \qquad \qquad &{} 0.2000 \qquad \qquad &{} 0.4000\\ \end{array} } \right] }\limits ^{{\begin{array}{*{20}l} &{} \mathcal {N}_1 &{}\qquad \qquad \mathcal {N}_2 &{}\quad \qquad \qquad \mathcal {N}_3 &{}\quad \mathcal {N}_4 \\ \end{array} }} \end{aligned}$$

So, the normalized form of weight vector is: $\mathcal {W}=\left( 0.2588,0.2575,0.2508,0.2329\right) ^T.$

Steps 9, 10.:

Utilizing the steps 9 and 10: CI value, maximum eigenvalue $\lambda _{\max }$ value, CR value, and RI value are computed as:

$$\begin{aligned} CI=-1, ~~~~~ \lambda _{\max }=1, ~~~~~ CR=-1.1111, ~~~~~ RI=0.9. \end{aligned}$$

Step 11. According to Eq. (23), the support degree $\mathbb {S}(\aleph _{\varrho \varsigma },\aleph _{{\varsigma }{k}})$ among ${\aleph }_{\varrho \varsigma }$ and $\aleph _{{\varrho }{k}}$ can be calculated. And $\mathbb {S}(\aleph _{\varrho \varsigma },\aleph _{{\varrho }{k}})$ can be represented as $\mathbb {S}_{\varsigma k}$, the results are shown as follows:

$$\begin{aligned} \mathbb {S}=\left[ \begin{array}{ccccccccccccccccccc} 1.0178 &{} 1.0363 &{} 1.0051 &{} 0.9980 &{} 0.9912 &{} 1.0162\\ 1.0127 &{} 0.9969 &{} 0.9914 &{} 1.0059 &{} 1.0356 &{} 1.0004\\ 1.0030 &{} 1.0005 &{} 1.0000 &{} 0.9997 &{} 1.0001 &{} 0.9965\\ 0.9989 &{} 0.9982 &{} 0.9975 &{} 0.9601 &{} 0.9462 &{} 0.9801\\ 0.9435 &{} 0.9744 &{} 0.8134 &{} 0.9974 &{} 0.9928 &{} 0.9745\\ 1.0329 &{} 1.0183 &{} 1.0298 &{} 0.9863 &{} 0.9952 &{} 0.9994\\ 1.0000 &{} 0.9982 &{} 1.0002 &{} 0.9882 &{} 0.9403 &{} 0.9521\\ 0.9971 &{} 0.9975 &{} 0.9996 &{} 1.0056 &{} 1.0016 &{} 0.9990\\ 0.9875 &{} 1.0123 &{} 1.0160 &{} 1.0722 &{} 1.0553 &{} 1.0040\\ \end{array} \right] \end{aligned}$$

Step 12. According to Eq. (24), the collective support matrix $\mathbb {T}(\aleph _{\varrho \varsigma })$ can be computed. And $\mathbb {T}(\aleph _{\varrho \varsigma })$ can be represented as $\mathbb {T}_{\varrho \varsigma }$, the results are shown as:

$$\begin{aligned} \mathbb {T}= \begin{array}{*{20}c} \mathcal {F}_1\\ \mathcal {F}_2\\ \mathcal {F}_3\\ \mathcal {F}_4\\ \mathcal {F}_5\\ \mathcal {F}_6\\ \mathcal {F}_7\\ \mathcal {F}_8\\ \mathcal {F}_9\\ \end{array} \mathop {\left[ {\begin{array}{*{20}l} 3.0592 \qquad \qquad &{} 3.0070 \qquad \qquad &{} 3.0505 \qquad \qquad &{} 3.0125\\ 3.0011 \qquad \qquad &{} 3.0543 \qquad \qquad &{} 3.0033 \qquad \qquad &{} 3.0274\\ 3.0035 \qquad \qquad &{} 3.0027 \qquad \qquad &{} 2.9967 \qquad \qquad &{} 2.9966\\ 2.9946 \qquad \qquad &{} 2.9052 \qquad \qquad &{} 2.9385 \qquad \qquad &{} 2.9239\\ 2.7313 \qquad \qquad &{} 2.9337 \qquad \qquad &{} 2.9462 \qquad \qquad &{} 2.7807\\ 3.0810 \qquad \qquad &{} 3.0144 \qquad \qquad &{} 3.0040 \qquad \qquad &{} 3.0244\\ 2.9984 \qquad \qquad &{} 2.9285 \qquad \qquad &{} 2.9385 \qquad \qquad &{} 2.8926\\ 2.9942 \qquad \qquad &{} 3.0043 \qquad \qquad &{} 3.0022 \qquad \qquad &{} 3.0002\\ 3.0157 \qquad \qquad &{} 3.1149 \qquad \qquad &{} 3.0884 \qquad \qquad &{} 3.0752\\ \end{array} } \right] }\limits ^{{\begin{array}{*{20}l} &{} \mathcal {N}_1 &{}\qquad \qquad \mathcal {N}_2 &{}\quad \qquad \qquad \mathcal {N}_3 &{}\quad \mathcal {N}_4 \\ \end{array} }} \end{aligned}$$

Step 13. According to Eq. (25), the results of power weight matrix are shown as follows:

$$\begin{aligned} \gamma = \begin{array}{*{20}c} \mathcal {F}_1\\ \mathcal {F}_2\\ \mathcal {F}_3\\ \mathcal {F}_4\\ \mathcal {F}_5\\ \mathcal {F}_6\\ \mathcal {F}_7\\ \mathcal {F}_8\\ \mathcal {F}_9\\ \end{array} \mathop {\left[ {\begin{array}{*{20}l} 0.2605 \qquad \qquad &{} 0.2559 \qquad \qquad &{} 0.2519 \qquad \qquad &{} 0.2317\\ 0.2575 \qquad \qquad &{} 0.2596 \qquad \qquad &{} 0.2497 \qquad \qquad &{} 0.2332\\ 0.2590 \qquad \qquad &{} 0.2577 \qquad \qquad &{} 0.2506 \qquad \qquad &{} 0.2327\\ 0.2623 \qquad \qquad &{} 0.2552 \qquad \qquad &{} 0.2506 \qquad \qquad &{} 0.2319\\ 0.2509 \qquad \qquad &{} 0.2632 \qquad \qquad &{} 0.2571 \qquad \qquad &{} 0.2288\\ 0.2620 \qquad \qquad &{} 0.2564 \qquad \qquad &{} 0.2491 \qquad \qquad &{} 0.2325\\ 0.2626 \qquad \qquad &{} 0.2567 \qquad \qquad &{} 0.2507 \qquad \qquad &{} 0.2301\\ 0.2584 \qquad \qquad &{} 0.2578 \qquad \qquad &{} 0.2509 \qquad \qquad &{} 0.2329\\ 0.2551 \qquad \qquad &{} 0.2601 \qquad \qquad &{} 0.2517 \qquad \qquad &{} 0.2330\\ \end{array} } \right] }\limits ^{{\begin{array}{*{20}l} &{} \mathcal {N}_1 &{}\qquad \qquad \mathcal {N}_2 &{}\qquad \qquad \quad \mathcal {N}_3 &{}\qquad \qquad \mathcal {N}_4 \\ \end{array} }} \end{aligned}$$

Step 14.:

By utilizing the PLT-SFWPA operator mentioned in Eq. (13), the overall assessment value for IoT applications ${\mathcal {F}_{\varrho }}(\varrho =1, 2,\ldots , 9)$ can be obtained as:

$$\begin{aligned} \begin{aligned} {\mathcal {F}_{1}}&=\langle \{\flat _{ 0.5668},(0.0000) \flat _{ 1.2947},(0.0000) \flat _{ 4.0000},(0.0000)\}, \\&\quad \{\flat _{-4.0000 },( 0.0000) \flat _{ -1.1834},(0.0000) \flat _{1.7786},(0.0000)\},\\&\quad \{ \flat _{-4.0000},(0.0000)\flat _{-4.0000},(0.0000) \flat _{ 0.6404},(0.0007)\}\rangle \\ {\mathcal {F}_{2}}&=\langle \{\flat _{-0.1455},(0.0000) \flat _{ 1.2816},(0.0000) \flat _{ 4.0000},(0.0000)\}, \\&\quad \{\flat _{-4.0000 },( 0.0000) \flat _{ -4.0000},(0.0000) \flat _{1.4805},(0.0000)\},\\&\quad \{ \flat _{-4.0000},(0.0000)\flat _{-1.1246},(0.0000) \flat _{ 0.8397},(0.0001)\}\rangle \\ {\mathcal {F}_{3}}&=\langle \{\flat _{-0.1542},(0.0000) \flat _{ 0.8517},(0.0000) \flat _{ 4.0000},(0.0000)\}, \\&\quad \{\flat _{-4.0000 },( 0.0000) \flat _{ -4.0000},(0.0000) \flat _{0.6175},(0.0000)\},\\&\quad \{ \flat _{-4.0000},(0.0000)\flat _{-1.9765},(0.0000) \flat _{ 1.0786},(0.0000)\}\rangle \\ {\mathcal {F}_{4}}&=\langle \{\flat _{-1.1655},(0.0000) \flat _{ -0.1410},(0.0000) \flat _{ 2.2074},(0.0000)\}, \\&\quad \{\flat _{-4.0000 },( 0.0000) \flat _{ -0.2343},(0.0000) \flat _{1.6267},(0.0000)\},\\&\quad \{ \flat _{-4.0000},(0.0000)\flat _{-1.5184},(0.0000) \flat _{ 1.0811},(0.0000)\}\rangle \\ {\mathcal {F}_{5}}&=\langle \{\flat _{-1.3839},(0.0000) \flat _{ -0.2889},(0.0000) \flat _{ 2.2678},(0.0000)\},\\&\quad \{\flat _{-1.8179 },( 0.0000) \flat _{ -0.2133},(0.0002) \flat _{2.2653},(0.0000)\},\\&\quad \{ \flat _{-4.0000},(0.0000)\flat _{-4.0000},(0.0000) \flat _{ 1.1743},(0.0000)\}\rangle \\ {\mathcal {F}_{6}}&=\langle \{\flat _{-1.5361},(0.0000) \flat _{ 0.1805},(0.0000) \flat _{ 2.1773},(0.0001)\}, \\&\quad \{\flat _{-4.0000 },( 0.0000) \flat _{ -0.8110},(0.0000) \flat _{1.8702},(0.0000)\},\\&\quad \{ \flat _{-4.0000},(0.0000)\flat _{-0.4554},(0.0000) \flat _{ 1.2984},(0.0000)\}\rangle \\ {\mathcal {F}_{7}}&=\langle \{\flat _{-0.5959},(0.0000) \flat _{ 0.7878},(0.0000) \flat _{ 4.0000},(0.0000)\}, \\&\quad \{\flat _{-4.0000 },( 0.0000) \flat _{ -0.9278},(0.0000) \flat _{1.5722},(0.0000)\},\\&\quad \{ \flat _{-4.0000},(0.0000)\flat _{-1.2650},(0.0000) \flat _{ 0.1574},(0.0000)\}\rangle \\ {\mathcal {F}_{8}}&=\langle \{\flat _{-0.6315},(0.0000) \flat _{ 0.7054},(0.0000) \flat _{ 2.4668},(0.0000)\}, \\&\quad \{\flat _{-4.0000 },( 0.0000) \flat _{ -0.4414},(0.0000) \flat _{1.2807},(0.0000)\},\\&\quad \{ \flat _{-4.0000},(0.0000)\flat _{-0.9307},(0.0000) \flat _{ 1.7556},(0.0000)\}\rangle \\ {\mathcal {F}_{9}}&=\langle \{\flat _{-0.2188},(0.0000) \flat _{ 0.8203},(0.0000) \flat _{ 4.0000},(0.0001)\}, \\&\quad \{\flat _{-4.0000 },( 0.0000) \flat _{ -1.3735},(0.0000) \flat _{0.9944},(0.0000)\},\\&\quad \{ \flat _{-4.0000},(0.0000)\flat _{-1.4535},(0.0000) \flat _{ 0.9814},(0.0000)\}\rangle \\ \end{aligned} \end{aligned}$$

Step 15.:

Eq. (5) is employed to establish the scores for the comprehensive evaluation values of the IoT applications as indicated in Table 13.

Table 13 The scores of 9 IoT applications utilizing the PLT-SFWPA operator

Step 16.:

By employing the results obtained from scores $\digamma ({\mathcal {F}_{\varrho }})(\varrho =1, 2,\ldots , 9)$, it becomes convenient to determine the ranking of the IoT applications ${\mathcal {F}_{\varrho }}(\varrho =1, 2,\ldots , 9)$. The IoT application with the highest value of $\digamma ({\mathcal {F}_{\varrho }})(\varrho =1, 2,\ldots , 9)$ is considered the best. Table 14 displays the ranking of the IoT applications.

Table 14 Ranking of nine IoT applications

Therefore, $\mathcal {F}_9$ is the best application.