500, 229258 (2019). Res. I hate spam & you may opt out anytime: Privacy Policy. Although weighted mean generally behaves in a similar way as average mean, they do have some contradictory properties. In recent years, with the continuous development of PFSs, the BM operators are expanded to develop some Pythagorean fuzzy BM operators [25, 26] and related multi-criterion interactive decision-making methods. This article illustrated how to compute weighted means in the R programming language. Table 1: Example Data with Numeric Column, Weights & Group Indicator. Step 5. \right\}\), \(\gamma \left( {\widetilde{cu}_{{i,gj_{{0}} }}^{{}} ,\widetilde{cu}_{i,gj}^{{}} } \right)\) indicate the grey relational coefficient between \(\widetilde{cu}_{{i,gj_{{0}} }}^{{}} \in \widetilde{cu}_{{i,gj_{{0}} }}^{{}}\) and \(\widetilde{cu}_{i,gj}^{{}} \in \widetilde{cu}_{i,gj}^{{}}\), and satisfy. Phase 2: Generation of criteria interaction coefficients based on ratings: Grey relational analysis (GRA) is widely used because of its convenience and reliability in calculating correlations. International Journal of Computational Intelligence Systems He wants you to calculate the weighted mean from the following data: Step 1:In Excel, there is an inbuilt formula for calculating the products of the numbers and their sum, which is one of the steps in calculating the weighted mean. Obtain the comprehensive impact matrix. Int. Calculate the center factors and the causal factors. Average is a method for calculating the central point of a given data set, and it is done using the traditional method of adding the numbers anddividing the sumby the number of data sets present. (2629), which are shown in Table 10. Support Syst. Data values with high weights contribute to the more weighted mean than the weights with lower weighted mean. Thus, if one performs poorly in chapter tests but does well in final exams, the weighted average of the grades will be relatively high. : AHP integrated TOPSIS and VIKOR methods with Pythagorean fuzzy sets to prioritize risks in self-driving vehicles. Appl. J. When the sum of all the weights adds to 1, then multiply each weight by its corresponding value and sum it all up. The above analysis shows that the existing operators have problems in setting the interaction weight \(w_{i,j}\). The CEO of a company has decided that he will continue the business only if the return on capital is more than the + [Cost of Debt * % of Debt * (1-Tax Rate)] url=https://www.wallstreetmojo.com/weighted-average-cost-capital-wacc/]weighted average cost of capitalWeightedThe. Syst. \(\frac{{\Delta_{{x_{i} x_{j} }} }}{{\Delta_{{x_{i} }} }}\) and \(\frac{{\Delta_{{x_{i} x_{j} }} }}{{\Theta_{{x_{i} }} }}\) are monotonically increasing with respect to \(\Delta_{ij}\). Step 1: Assign a weight to each value in a given data set. Step 3. Get regular updates on the latest tutorials, offers & news at Statistics Globe. Step 4. If you accept this notice, your choice will be saved and the page will refresh. Int J Comput Intell Syst 15, 94 (2022). $$, $$ \widetilde{D} = \left( {\widetilde{d}_{g}^{i} } \right)_{{u_{i} \times m}} ,{\text{where}}\quad \widetilde{d}_{g}^{i} = \mu_{i,g}^{2} - v_{i,g}^{2} . 512, 14811502 (2020), Fernndez, E., Navarro, J., Solares, E.: A hierarchical interval outranking approach with interacting criteria. It sums up the numbers and divides them with the count of numbers which provides us with the mean. \[\sum_{i=1}^{4}\]w\[_{i}\] = w\[_{1}\] + w\[_{2}\] + w\[_{3}\]. $$, \(N_{d} \left( {x_{i} } \right) = N_{d} \left( {x_{0} } \right)\), $$ \begin{aligned} {\text{INWIBM}}_{{{\text{dual}}}}^{{^{p,q} }} \left( X \right) & = N_{d} \left( {{\text{INWIBM}}^{p,q} \left( {N_{d} \left( {x_{1} } \right),N_{d} \left( {x_{1} } \right), \ldots ,N_{d} \left( {x_{n} } \right)} \right)} \right) \\ & = N_{d} \left( {N_{d} \left( {x_{0} } \right)} \right) = x_{0} \\ \end{aligned} $$, $$ \begin{aligned} & {\text{INWIBM}}_{{{\text{dual}}}}^{{^{p,q} }} \left( X \right) = N_{d} \left( {{\text{INWIBM}}^{p,q} \left( {N_{d} \left( {x_{1} } \right),N_{d} \left( {x_{1} } \right), \ldots ,N_{d} \left( {x_{n} } \right)} \right)} \right) \\ & \quad \le {\text{INWIBM}}_{{{\text{dual}}}}^{{^{p,q} }} \left( Y \right) = N_{d} \left( {{\text{INWIBM}}^{p,q} \left( {N_{d} \left( {y_{1} } \right),N_{d} \left( {y_{1} } \right), \ldots ,N_{d} \left( {y_{n} } \right)} \right)} \right) \\ & \quad \Leftrightarrow \\ & \quad {\text{INWIBM}}^{p,q} \left( {N_{d} \left( {x_{1} } \right),N_{d} \left( {x_{1} } \right), \ldots ,N_{d} \left( {x_{n} } \right)} \right) \\ & \quad \ge {\text{INWIBM}}^{p,q} \left( {N_{d} \left( {y_{1} } \right),N_{d} \left( {y_{1} } \right), \ldots ,N_{d} \left( {y_{n} } \right)} \right) \\ & \quad \Leftrightarrow \\ & \quad N_{d} \left( {x_{i} } \right) \ge N_{d} \left( {y_{i} } \right)\left( {i = 1,2, \ldots ,n} \right) \\ & \quad \Leftrightarrow \\ & \quad x_{i} \ge y_{i} \left( {i = 1,2, \ldots ,n} \right) \\ \end{aligned} $$, $$ {\text{INWIBM}}_{{{\text{dual}}}}^{p,q} \left( {x_{l} ,x_{l} , \ldots ,x_{l} } \right) = x_{l} ,{\text{INWIBM}}_{{{\text{dual}}}}^{p,q} \left( {x_{u} ,x_{u} , \ldots ,x_{u} } \right) = x_{u} $$, $$ \begin{gathered} {\text{INWIBM}}_{{{\text{dual}}}}^{p,q} \left( {x_{l} ,x_{l} , \ldots ,x_{l} } \right) \le {\text{INWIBM}}_{{{\text{dual}}}}^{p,q} \left( {x_{1} ,x_{2} , \ldots ,x_{n} } \right) \hfill \\ {\text{INWIBM}}_{{{\text{dual}}}}^{p,q} \left( {x_{1} ,x_{2} , \ldots ,x_{n} } \right) \le {\text{INWIBM}}_{{{\text{dual}}}}^{p,q} \left( {x_{u} ,x_{u} , \ldots ,x_{u} } \right). 4.1, an online multi-dimensional rating aggregation decision-making approach is developed. weighted_columns <- numeric() J. Appl. Now, we can calculate the weighted mean with the following R code: data %>% # Weighted mean by group
Copyright Statistics Globe Legal Notice & Privacy Policy. In this paper, a data-driven interaction operator method to generate interaction coefficients is used in the context of user rating. : Online-review analysis based large-scale group decision-making for determining passenger demands and evaluating passenger satisfaction: case study of high-speed rail system in China. Cite this article. Appl. $$, \(D = \left( {d_{g}^{i} } \right)_{{u_{i} \times m}}\), $$ d_{g}^{i} = \left( {\mu_{i,g} ,v_{i,g} } \right) = {\text{PFWIBM}}\left( {\widetilde{cu}_{g}^{i,1} ,\widetilde{cu}_{g}^{i,2} , \ldots ,\widetilde{cu}_{g}^{i,n} } \right). J. Intell. Due to the large amount of data, the information matrix \(CU = \left( {cu_{g}^{i,j} } \right)_{{u_{i} \times 8}}\)\(\left( {i = 1,2, \ldots ,5;\,\,j = 1,2, \ldots ,8} \right)\) is not provided in detail. $$, \(X = \{ x_{i} \left| {i = 1,2, \ldots ,n} \right.\} \in \left[ {0,1} \right]^{n}\), $$ {\text{NWIBM}}^{p,q} \left( X \right) = \left( {\sum\limits_{i = 1}^{n} {w_{i} x_{i}^{p} \left( {\sum\limits_{j = 1}^{n} {w_{i,j} x_{j}^{q} } \bigg/\sum\limits_{j = 1}^{n} {w_{i,j} } } \right)} } \right)^{{\frac{1}{p + q}}} , $$, \(w_{i,j} \in [0,1]\,\,\left( {i \ne j;\,\, i,j = 1,2, \ldots ,n} \right)\), \(w_{i,i} = 0\left( {i = 1,2, \ldots n} \right)\), \(w_{i} \in [0,1]\left( {i = 1,2, \ldots ,n} \right)\), \(\sum\nolimits_{i = 1}^{n} {w_{i} = 1}\), \(w_{i,j} \in [0,1]\left( {i \ne j;i,j = 1,2,,n} \right)\), $$ {\text{NWBM}}^{p,q} \left( X \right) = \left( {\sum\limits_{i = 1}^{n} {w_{i} x_{i}^{p} \left( {\sum\limits_{j = 1,i \ne j}^{n} {\frac{{w_{j} x_{j}^{q} }}{{1 - w_{i} }}} } \right)} } \right)^{{\frac{1}{p + q}}} . J. Oper. Int. =73+378+459+90 =1000 Divide the numerator by the denominator =2566/1000 =2.566 Comput. $$, $$ {\text{INWIBM}}^{p,q} \left( {\mu_{1} ,\mu_{2} , \ldots \mu_{n} } \right) = {\text{INWIBM}}^{p,q} \left( {\mu_{0} ,\mu_{0} , \ldots \mu_{0} } \right) = \mu_{0} $$, $$ {\text{INWIBM}}_{{{\text{dual}}}}^{{^{p,q} }} \left( {v_{1} ,v_{2} , \ldots ,v_{n} } \right) = {\text{INWIBM}}_{{{\text{dual}}}}^{{^{p,q} }} \left( {v_{0} ,v_{0} , \ldots ,v_{0} } \right) = v_{0} . Intell. Int. Step 6. For simplicity, we call the two tuples \(\left( {\mu_{P} \left( x \right), \, v_{P} \left( x \right)} \right)\) as Pythagorean fuzzy number (PFN) and simply express it as \(P = \left( {\mu_{P} ,v_{P} } \right)\). Arithmetic mean denotes the average of all the observations of a data series. 292, 126047 (2021), Xiao, L., Huang, G., Pedrycz, W., et al. This tutorial explains how to compute the weighted mean in the R programming language. Comparison of spaces of the PFNs and IFNs, [25] Let \(\alpha = \left( {\mu_{\alpha } ,v_{\alpha } } \right)\) be an PFN, the score function of \(\alpha\) be defined as. CFA Institute Does Not Endorse, Promote, Or Warrant The Accuracy Or Quality Of WallStreetMojo. Soft Comput. Let \(\alpha_{i} = \left( {\mu_{i} ,v_{i} } \right)\left( {i = 1,2, \ldots ,n} \right)\) be a collection of PFNs, and \(p,q \ge 0\), the PFWIBM operator is defined as follows: where \(w_{i,j} \ge 0\left( {i \ne j;i,j = 1,2, \ldots ,n} \right)\). Int. 357, 6187 (2016), Article The sum of rows and the sum of columns are separately denoted as D and R within the comprehensive impact matrix \(M\). Syst. Based on the criterion weights \(w_{{j_{0} }}^{c}\) and the interaction coefficients \(w_{{j_{0} j}}\), the Pythagorean fuzzy matrix \(\widetilde{CU}\) and the user comprehensive rating matrix \(\widetilde{D}\) are obtained by using the PFWIBM operator. group_by(group) %>%
Step 3. The interaction coefficients of the existing BM operators are often determined directly by experts. : Pythagorean hesitant fuzzy Choquet integral aggregation operators and their application to multi-attribute decision-making. where \(w_{i,j} \in [0,1]\left( {i \ne j;i,j = 1,2,,n} \right)\) be a collection of weights such that \(w_{i,j} \ge 0\) for all \(i,j = 1, \ldots ,n\) and \(w_{i,i} = 0\left( {i = 1,2,n} \right)\). \(\Delta_{{x_{i} x_{j} }} = \Delta_{ij} x_{j}\) monotonically increases with respect to interaction coefficient \(w_{i,j}\). 23(1), 251267 (2019), Chen, Z.S., Chin, K.S., Tsui, K.L. Based on the above definition, Zhang and Xu [27] defined the following concept of the distance measure for PFNs. Google Scholar, Chen, Z.S., Chin, K.S., Li, Y.L., et al. Of cause, this is also true for the computation of the weighted mean. By using our website, you agree to our use of cookies (, Calculation of Weighted Mean (Step by Step), + [Cost of Debt * % of Debt * (1-Tax Rate)] url=https://www.wallstreetmojo.com/weighted-average-cost-capital-wacc/]weighted average cost of capital. Clean. Accordingly, the improved NWIBM operator is defined as follows. The numbers 40, 45, 80, 75 and 10 have weights 1, 2, 3, 4, and 5 respectively. Multiply each number and the relevant weight assigned to that number (w1 by x1, w2 by x2, and so on). Step 1. : A heterogeneous QUALIFLEX method with criteria interaction for multi-criteria group decision making. Sci. Yi Yang. data <- data.frame(x1, w1, group) # Create data frame. Let It gives a businessman a better understanding of his expenses. Step 1.2 The comprehensive Pythagorean fuzzy matrix \(D = \left( {d_{g}^{i} } \right)_{{u_{i} \times m}}\) is obtained using the PFWIBM operator to aggregate \(\widetilde{CU} = \left( {\widetilde{cu}_{g}^{i,j} } \right)_{{u_{i} \times n}}\): Step 1.3 The user comprehensive rating matrix \(\widetilde{D}\) is obtained by Eq. Let \(p,q \ge 0\), and \(X = \{ x_{i} \left| {i = 1,2, \ldots ,n} \right.\} \in \left[ {0,1} \right]^{n}\) be a collection of crisp data, the improved NWIBM operator is given by. CJ, ZY and LL implemented the editing work. Correspondence to 11(1), 10911110 (2018), Akram, M., Wasim, F., Alcantud, J.C.R., et al. J. Intell. : A novel approach to three-way conflict analysis and resolution with Pythagorean fuzzy information. Weighted mean can aid an individual in making decisions where some attributes have more significance than others. Syst. : A novel VIKOR approach based on entropy and divergence measures of Pythagorean fuzzy sets to evaluate renewable energy technologies in India. J. Intell. \\ \end{aligned} $$, $$ r\_expert = \left( {re_{{j_{0} j}} } \right)_{n \times n} ,re_{{j_{0} j}} = \left\{ {\begin{array}{*{20}c} {0,} & {{\text{if }}j_{0} = j{\text{ or }}m_{{j_{0} j}} < \theta } \\ {m_{{j_{0} j}} ,} & {{\text{if }}m_{{j_{0} j}} \ge \theta } \\ \end{array} } \right.. $$, $$ w_{{j_{0} }}^{c} = \frac{{\sqrt {\left( {R_{{j_{0} }} + D_{{j_{0} }} } \right)^{2} + \left( {R_{{j_{0} }} - D_{{j_{0} }} } \right)^{2} } }}{{\sum\nolimits_{{j_{0} = 1}}^{n} {\sqrt {\left( {R_{{j_{0} }} + D_{{j_{0} }} } \right)^{2} + \left( {R_{{j_{0} }} - D_{{j_{0} }} } \right)^{2} } } }},\quad {\text{where}}\quad w_{{j_{0} }}^{c} \in \left[ {0,1} \right],\sum\limits_{{j_{0} = 1}}^{n} {w_{{j_{0} }}^{c} = 1} . Due to the complexity and uncertainty of objective things and the ambiguity of human thinking, the study of MCDM problems in uncertain environments has attracted great attention. 34(6), 13031336 (2019), Wang, L., Li, N.: Pythagorean fuzzy interaction power Bonferroni mean aggregation operators in multiple attribute decision making. Obtain the criteria interaction matrix \(r\_expert\) provided by the expert. J. Intell. Introduction to Hypothesis Testing | Theoretical Concepts & Example. Next, the effect of parameters \(p\) and \(q\) on the ranking results are observed, and the ratings of the alternatives using different values of \(p\) and \(q\) in Step 5 of Sect. https://doi.org/10.1109/TFUZZ.2022.3179594, Demir, U.Y., Diner, H., Yksel, S., et al. : Power-average-operator-based hybrid multiattribute online product recommendation model for consumer decision-making. $$, \(\alpha^{ + } = \left( {\max_{j} \left\{ {\mu_{j} } \right\},\min_{j} \left\{ {v_{j} } \right\}} \right)\), \(\alpha^{ - } = \left( {\min_{j} \left\{ {\mu_{j} } \right\},\max_{j} \left\{ {v_{j} } \right\}} \right)\), $$ \alpha^{ - } \le {\text{PFWIBM}}^{p,q} \left( {\alpha_{i} } \right) \le \alpha^{ + } . Therefore, the NWIBM operator satisfies boundedness, idempotency and monotonicity. Combined with the above analysis, a GRA method based on PFSs [34] is proposed. Step 2. Consumers can use the public information platforms to make initial screening before purchasing products and obtain reference information to provide decision reserves for offline store selection. Appl. Weighted mean is the mean where some values contribute more than others. [25] Let \(p = \left( {\mu ,v} \right),p_{1} = \left( {\mu_{1} ,v_{1} } \right)\) and \(p_{2} = \left( {\mu_{2} ,v_{2} } \right)\) be three PFNs, then we have: The desirable characteristic of the Bonferroni mean (BM) [20] is that it can capture the interrelationship between input arguments.