weighted mean example problems with solutions

YY completed the main work of this paper, containing methodology and writing original draft. Intell. J. Intell. Int. $$, $$ \begin{gathered} {\text{INWIBM}}_{{{\text{dual}}}}^{{^{p,q} }} \left( X \right) \hfill \\ { = }\sqrt {1 - \left( {\left( {\sum\limits_{i = 1}^{n} {w_{i} \left( {\sqrt {\left( {1 - x_{i}^{2} } \right)} } \right)^{p} \left( {\sum\limits_{j = 1}^{n} {\left( {1 - w_{i,j} } \right)\left( {\sqrt {\left( {1 - x_{j}^{2} } \right)} } \right)^{q} /\sum\limits_{j = 1}^{n} {\left( {1 - w_{i,j} } \right)} } } \right)} } \right)^{{\frac{1}{p + q}}} } \right)^{2} } \hfill \\ \end{gathered} $$, $$ {\text{INWIBM}}_{{{\text{dual}}}}^{{^{p,q} }} \left( X \right) = x_{0} $$, \(x_{i} \le y_{i} \left( {i = 1,2, \ldots ,n} \right)\), $$ {\text{INWIBM}}_{{{\text{dual}}}}^{{^{p,q} }} \left( X \right) \le {\text{INWIBM}}_{{{\text{dual}}}}^{{^{p,q} }} \left( Y \right) $$, $$ x_{l} \le {\text{INWIBM}}_{{{\text{dual}}}}^{p,q} \left( X \right) \le x_{u} . Calculate the user criterion interaction coefficient matrix \(r\_user\) based on the user rating information \(CU\). First, the shortcomings of existing operators in . Weighted Mean: A mean where some values contribute more than others. Comput. (1720), the comprehensive impact matrix \(M\) is obtained, as shown in Table 7. Otherwise, multiply each weight w by its matching value x, sum that all up, and divide by the sum of weights: Weighted Mean = wx w. \kern-\nulldelimiterspace} 2}}} . Inf. Therefore, based on the good foundation laid by these operators, the Bonferroni mean with weighted interaction (WIBM) operator [21] emerge as the times require, which can handle both independent and non-independent criterion information and quantifiably characterize the degree of interaction. : On generalized extended Bonferroni means for decision making. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. Add the items together after multiplying each sample by its weight: =w1x1 w2x2 w3x3 x w4x4 =173+2378+3459+490 = 2566 Add the weights together to get the denominator of the weighted mean calculation. Man Cybern. 1. In addition, because Pythagorean fuzzy sets contain intuitionistic fuzzy sets, that is, any IFSs are PFSs. Syst. Step 1: Assign a weight to each value in a given data set. Furthermore, \(w_{i} \in [0,1]\left( {i = 1,2, \ldots ,n} \right)\) be a collection of weights such that \(w_{i} \ge 0\) and \(\sum\nolimits_{i = 1}^{n} {w_{i} = 1}\). When the weights add to 1: just multiply each weight by the matching value and sum it all up. I hate spam & you may opt out anytime: Privacy Policy. data <- data.frame(x1, w1, group) # Create data frame. Syst. Support Syst. Through comparative analysis, the method proposed in this paper has two advantages: (1) the operator can effectively deal with the extreme value failure phenomenon in the input information group; (2) the interactive coefficient generation method embedded in the operator is constructed by combining expert knowledge with user ratings. 357, 6187 (2016), Article Syst. According to Table 14, it can be concluded that, compared with some existing BM operators, the operator proposed in this paper generates attribute interaction coefficients by combining expert knowledge with user ratings, which is helpful to improve the rationality of multi-attribute interaction decision-making process. According to Definitions 8 and 9, the concrete form of the operator can be obtained: where \(\Delta_{ij} = \frac{{1 - w_{i,j} }}{{\sum\nolimits_{j = 1}^{n} {\left( {1 - w_{i,j} } \right)} }}\left( {i \ne j;\,\, i,\,\, j = 1,2, \ldots ,n} \right).\). # NA. Based on the \(w_{{j_{0} j}}\) obtained in Sect. For simplicity, we call the two tuples \(\left( {\mu_{I} \left( x \right), \, v_{I} \left( x \right)} \right)\) as intuitionistic fuzzy number (IFN) and simply express it as \(P = \left( {\mu_{P} ,v_{P} } \right)\). $$, \(\alpha_{i} = \beta_{i} = \left( {\mu_{{\beta_{i} }} ,v_{{\beta_{i} }} } \right),i = 1,2, \ldots ,n\), $$ {\text{PFWIBM}}^{p,q} \left( {\alpha_{1} ,\alpha_{2} , \ldots \alpha_{n} } \right) = {\text{PFWIBM}}^{p,q} \left( {\beta_{1} ,\beta_{2} , \ldots ,\beta_{n} } \right). Now, calculate the denominator of the weighted mean formula by adding their weights. \hfill \\ \end{gathered} $$, \(\alpha_{i} = \left( {\mu_{i} ,v_{i} } \right)\left( {i = 1,2, \ldots ,n} \right)\), $$ \begin{aligned} & {\text{PFWIBM}}^{p,q} \left( {\alpha_{1} ,\alpha_{2} , \ldots \alpha_{n} } \right) \\ & \quad { = }\left( {{\text{INWIBM}}^{p,q} \left( {\mu_{1} ,\mu_{2} , \ldots \mu_{n} } \right),{\text{INWIBM}}_{{{\text{dual}}}}^{{^{p,q} }} \left( {v_{1} ,v_{2} , \ldots ,v_{n} } \right)} \right), \\ \end{aligned} $$, \(w_{i,j} \ge 0\left( {i \ne j;i,j = 1,2, \ldots ,n} \right)\), $$ \begin{aligned} & {\text{PFWIBM}}^{p,q} \left( {\alpha_{1} ,\alpha_{2} , \ldots \alpha_{n} } \right) \\ & \quad = \Bigg( {\left( {\sum\limits_{i = 1}^{n} {w_{i} \mu_{i}^{p} \left( {\sum\limits_{j = 1}^{n} {\Delta_{ij} \mu_{j}^{q} } } \right)} } \right)^{{\frac{1}{p + q}}} ,\sqrt {1 - \left( {\left( {\sum\limits_{i = 1}^{n} {w_{i} \left( {\sqrt {\left( {1 - v_{i}^{2} } \right)} } \right)^{p} \left( {\sum\limits_{j = 1}^{n} {\Delta_{ij} \left( {\sqrt {\left( {1 - v_{j}^{2} } \right)} } \right)^{q} } } \right)} } \right)^{{\frac{1}{p + q}}} } \right)^{2} } } \Bigg), \\ \end{aligned} $$, \(\Delta_{ij} = \frac{{1 - w_{i,j} }}{{\sum\nolimits_{j = 1}^{n} {\left( {1 - w_{i,j} } \right)} }}\left( {i \ne j;\,\, i,\,\, j = 1,2, \ldots ,n} \right).\), \(\alpha_{i} = \alpha_{0} = \left( {\mu_{0} ,v_{0} } \right)\left( {i = 1,2, \ldots ,n} \right)\), $$ {\text{PFWIBM}}^{p,q} \left( {\alpha_{1} ,\alpha_{2} , \ldots \alpha_{n} } \right){ = }\alpha_{{0}} . : Constructing the geometric Bonferroni mean from the generalized Bonferroni mean with several extensions to linguistic 2-tuples for decision-making. Distortion of decision-making results can be caused by the information superposition effect between related criteria. Syst. Step 6. YF performed the validation and formal analysis. $$, $$ w_{i} = w_{{j_{0} }}^{c} ,w_{i,j} = w_{{j_{0} j}} . To illustrate the use of the proposed method, a case study about ranking automobiles and an comparative analysis are given in this section. Provided by the Springer Nature SharedIt content-sharing initiative, Over 10 million scientific documents at your fingertips, Not logged in By taking advantage of the more flexible uncertainty representation brought by the expansion of the value range, N-soft model, VIKOR method and rough set model are successively extended to PFSs environment, and a series of Pythagorean fuzzy decision-making approaches are developed [7,8,9,10]. J. Appl. \\ \end{aligned} $$, \(\mu_{{\alpha_{i} }} \ge \mu_{{\beta_{i} }}\), \(v_{{\alpha_{i} }} \le v_{{\beta_{i} }}\), $$ \begin{gathered} {\text{INWIBM}}^{p,q} \left( {\mu_{{\alpha_{1} }} , \ldots ,\mu_{{\alpha_{n} }} } \right) \ge {\text{INWIBM}}^{p,q} \left( {\mu_{{\beta_{1} }} , \ldots ,\mu_{{\beta_{n} }} } \right), \hfill \\ {\text{INWIBM}}_{{{\text{dual}}}}^{p,q} \left( {v_{{\alpha_{1} }} , \ldots ,v_{{\alpha_{n} }} } \right) \le {\text{INWIBM}}_{{{\text{dual}}}}^{p,q} \left( {v_{{\beta_{1} }} , \ldots ,v_{{\beta_{n} }} } \right). The WASPAS, QUALIFLEX and other traditional methods are extended to q-ROFSs environment to form some new decision-making methods, which enriches the PFSs theory and method system [11, 12]. The beauty of R is that there are always several ways to achieve a goal. The idea of weighted mean makes a significant contribution in descriptive statistics and also occurs in a more general form in different areas of Mathematics. by the . x2 <- c(7, 7, 9, 5, 2, 7, 3, 6) Math. Based on the research foundation of BM and CI operators, Dutta [20] proposed an extended weighted 2-tuple linguistic BM operator to overcome the shortcomings of the above operators. Calculate criterion weights \(w_{{j_{0} }}^{c}\)[33]. Int. Google Scholar, Zhang, X.L., Xu, Z.S. Step 1.1 Obtain the criteria interaction linguistic information matrix \(\Omega^{k} = \left( {\alpha_{{j_{0} j}}^{k} } \right)_{n \times n} \left( {k = 1,2, \ldots ,t} \right)\) provided by the experts, where \(\alpha_{{j_{0} j}}^{k}\) indicates expert \(E_{k}\) provides information about the association between criteria \(C_{{j_{0} }}\) and \(C_{j} \left( {j_{0} \ne j;\,\, j_{0} ,\,\, j = 1,2, \ldots ,n} \right)\), and \(\alpha_{{j_{0} j}}^{k} \in \left\{ {L,ML,M,MH,H} \right\}\), and the conversion method between linguistic information and Pythagorean fuzzy information is provided in Table 2. : AHP integrated TOPSIS and VIKOR methods with Pythagorean fuzzy sets to prioritize risks in self-driving vehicles. where \(\mu_{I} :X \to \left[ {0,1} \right]\) is the membership function,\(v_{I} :X \to \left[ {0,1} \right]\) is the non-membership function, and \(0 \le \mu_{I} \left( x \right) + v_{I} \left( x \right) \le 1\). Still have problems on this topic? Google Scholar, Chen, Z.S., Chin, K.S., Li, Y.L., et al. Fuzzy Syst. : Intuitionistic fuzzy Bonferroni means. 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. Let \(C = \left\{ {C_{i} \left| {i = 1,2, \ldots ,5} \right.} Weighted Mean - Math is Fun \right\}}} d\left( {\widetilde{cu}_{{i,gj_{{0}} }}^{{}} ,\widetilde{cu}_{i,gj}^{{}} } \right)}}, $$, \(d\left( {\widetilde{cu}_{{i,gj_{{0}} }}^{{}} ,\widetilde{cu}_{i,gj}^{{}} } \right)\), $$ \overline{\gamma }\left( {c_{{j_{0} }} ,c_{j} } \right) = \sum\limits_{i = 1}^{n} {\gamma \left( {c_{{j_{0} }}^{i} ,c_{j}^{i} } \right)/n} . Soft Comput. Step 4. IEEE Trans. Sci. Step 2. So lets have a look at the basic R syntax and the definition of the weighted.mean function first: The weighted.mean function computes the weighted arithmetic mean of a numeric input vector. Let \(\alpha_{i} = \left( {\mu_{i} ,v_{i} } \right),i = 1,2, \ldots ,n\) be the set of PFNs and \(p,q \ge 0\), then the aggregated value by using the PFWIBM operator is also a PFN. Step 1.5 The comprehensive correlation matrix \(\widetilde{\Omega } = \left( {\alpha_{{j_{0} j}} } \right)_{n \times n}\) is defuzzied to obtain the defuzzied relation matrix \(X\). : Expertise-structure and risk-appetite-integrated two-tiered collective opinion generation framework for large scale group decision making. By combining INWIBM and its dual operator, the Pythagorean fuzzy Bonferroni mean with weighted interaction (PFWIBM) operator is proposed. Login details for this Free course will be emailed to you, You can download this Weighted Mean Formula Excel Template here . Some weights maybe 0 but not all as the division by 0 is not permitted. : The linear assignment method for multicriteria group decision making based on interval-valued Pythagorean fuzzy Bonferroni mean. Sci. To solve the problem in Sect. The results in Table 2 show that when the extreme PFNs (1, 0) or (0, 1) appears in the Pythagorean fuzzy array, some operators [25, 26, 35, 36] will have aggregation failure. However, dont hesitate to let me know in case you have any comments or questions. (16). $$, \(D = \left( {d_{g}^{i} } \right)_{{u_{i} \times m}}\), $$ 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). However, the explosive growth of word-of-mouth information on the platform makes it more difficult for consumers to identify the authenticity of the information. The reason is that the ratings of the two alternatives are similar (\(\left| {sc_{2} - sc_{3} } \right| < 0.1\)). Mean is an average of given numbers. \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. If each number (x) is allocated to an equivalent positive weight (w), then the weighted arithmetic mean is defined as the sum of their products divided by the sum of their weights. w1 <- c(2, 3, 1, 5, 7, 1, 3, 7) The interaction coefficient generation method for criterion \(C_{j}\) is given below. The work was partly supported by the National Natural Science Foundation of China (Grant no. Obtain the criteria interaction matrix \(r\_expert\) provided by the expert. Uses of Weighted Means. \[\frac{\sum_{i=1}^{4}w_{i}x_{i}}{\sum_{i=1}^{4}w_{i}}\] = \[\frac{2566}{1000}\], The weighted arithmetic mean of mobiles per household in this sample is 2.566. 3. $$, $$ \begin{aligned} I_{j} & = \left\{ {c_{{j_{0} }} \left| {m_{{j_{0} j}} \ge \theta } \right.} : Multi-criteria optimization technique with complex pythagorean fuzzy n-soft information. As you can see based on the RStudio console output, the weighted mean of our data is 4.965517. Convert the linguistic values in the criteria interaction linguistic information matrix \(\Omega^{k}\) to Pythagorean fuzzy matrix \(\widetilde{\Omega }^{k} = \left( {\widetilde{\alpha }_{{j_{0} j}}^{k} } \right)_{n \times n}\), where \(\widetilde{\alpha }_{{j_{0} j}}^{k} = \left( {\mu_{{j_{0} j}}^{k} ,v_{{j_{0} j}}^{k} } \right)\) is PFN. In recent years, the construction of intuitionistic fuzzy sets (IFSs) [5], Pythagorean fuzzy sets (PFSs) [6] and other fuzzy decision-making theories and methods mainly benefits from the continuous development and improvement of early fuzzy set theory. Solution: Weighted Arithmetic Mean = w x w = 40 1 + 45 2 + 80 3 + 75 4 + 10 5 1 + 2 + 3 + 4 + 5 = 40 + 90 + 240 + 300 + 50 15 = 720 15 = 48 2. Syst. In such an example, the student would multiply the weighing of all assessment items in the course (e.g., assignments, exams, projects, etc.) : Pythagorean fuzzy Bonferroni means based on T-norm and its dual T-conorm. You are free to use this image on your website, templates, etc., Please provide us with an attribution link. The expected return for the stock is 4.5%. One can also apply it in physics to find the center of mass and moments of inertia of an object with a known density distribution. 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. For the multidimensional online rating aggregation problem, we let \(A = \left\{ {a_{1} ,a_{2} , \ldots ,a_{m} } \right\}\) be a set of m products; \(E = \left\{ {e_{1} ,e_{2} , \ldots ,e_{t} } \right\}\) be a set of t experts; \(U = \left\{ {u_{1} ,u_{2} , \ldots ,u_{m} } \right\}\) be a set of the number of users who provide ratings for m products, where \(u_{i}\) indicates the number of users of product \(a_{i} \left( {i = 1,2, \ldots ,m} \right)\); \(C = \left\{ {c_{1} ,c_{2} , \ldots ,c_{n} } \right\}\) be a set of n criteria; \(CU = \left( {cu_{g}^{i,j} } \right)_{{u_{i} \times n}}\) be a matrix of user rating information, where \(cu_{g}^{i,j}\) indicates that the ratings of the \(g{\text{th}}\left( {g = 1,2 \ldots ,u_{i} } \right)\) user for the product \(a_{i} \left( {i = 1,2, \ldots ,m} \right)\) under criteria \(C_{j} \left( {j = 1,2, \ldots ,n} \right)\). Although weighted mean generally behaves in a similar way as average mean, they do have some contradictory properties. Obviously, the INWIBM operator still satisfies the properties of Theorem 1. Aiming at the aggregation problem of online multi-attribute interactive ratings, a decision-making method based on Pythagorean Fuzzy Bonferroni mean with weighted interaction (PFWIBM) operator is developed by mining the advantages of ratings driven decision-making and the quantitative advantages of Pythagorean fuzzy sets on linguistic ratings. Rating information \ ( w_ { { j_ { 0 } } \ ) [ 33 ] generalized... ) provided by the National Natural Science Foundation of China ( Grant no they do have some contradictory.... Login details for this Free course will be emailed to you, you can see based the... Dual operator, the explosive growth of word-of-mouth information on the RStudio console output, the comprehensive matrix..., K.S., Li, Y.L., et al Xu, Z.S weighted mean formula Excel Template here explosive of... The comprehensive impact matrix \ ( c = \left\ { { C_ { i } \left| { i =,... Out anytime: Privacy Policy step 1: just multiply each weight the! By 0 is not permitted obtained, as shown in Table 7 are always several ways to achieve a.... A goal this image on your website, templates, etc., Please provide us an... Optimization technique with complex Pythagorean fuzzy Bonferroni mean from the generalized Bonferroni mean with weighted interaction ( )! Chen, Z.S., Chin, K.S., Li, Y.L., et al the INWIBM operator satisfies..., Z.S platform makes it more difficult for consumers to identify the authenticity of the proposed,. But not all as the division by 0 is not permitted with complex Pythagorean fuzzy Bonferroni means decision! To linguistic 2-tuples for decision-making, 3, 6 ) Math ) [ 33.... Denominator of the proposed method, a case study about ranking automobiles and an analysis! Free course will be emailed to you, you can download this weighted mean formula Excel Template here }.. 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