: Power-average-operator-based hybrid multiattribute online product recommendation model for consumer decision-making. 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. and then we can apply the weightedMean R function as follows: weightedMean(x1, w1) # Apply weightedMean function
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. Fuzzy Syst. However, the operation mechanism of the PFWIBM is the focus of the decision-making model designed in this paper, and the division and processing of online ratings are not involved. All authors read and approved the final manuscript. Calculate the user criterion interaction coefficient matrix \(r\_user\) based on the user rating information \(CU\). For example, the degree of interaction between criterion \(c_{i}\) and \(c_{j}\), and whether criterion \(c_{i}\) and criterion \(c_{j}\) are independent, are subjectively decided by experts or decision-making makers. Step 2:Calculate the sum of the weights using the formula =SUM(B2: B5), where the range B2: B5 represents the weights. List the numbers and weights in tabular form. J. Intell. \kern-\nulldelimiterspace} n}\left( {i = 1,2, \ldots ,n} \right)\), $$ {\text{WBM}}^{p,q} \left( X \right) = \left( {\frac{1}{n}\sum\limits_{{i{ = }1}}^{n} {x_{i}^{p} } \frac{1}{n - 1}\left( {\sum\limits_{j = 1;j \ne i}^{n} {x_{j}^{q} } } \right)} \right)^{{\frac{1}{p + q}}} . Pythagorean Fuzzy Bonferroni Mean with Weighted Interaction Operator and Its Application in Fusion of Online Multidimensional Ratings, $$ I = \, \left\{ {\left\langle {x, \, \mu_{I} \left( x \right), \, v_{I} \left( x \right)} \right\rangle \left| {x \in } \right.X} \right\}, $$, \(0 \le \mu_{I} \left( x \right) + v_{I} \left( x \right) \le 1\), \(\left( {\mu_{I} \left( x \right), \, v_{I} \left( x \right)} \right)\), $$ P = \, \left\{ {\left\langle {x, \, \mu_{P} \left( x \right), \, v_{P} \left( x \right)} \right\rangle \left| {x \in } \right.X} \right\}, $$, \(0 \le \left( {\mu_{P} \left( x \right)} \right)^{2} + \left( {v_{P} \left( x \right)} \right)^{2} \le 1\), \(\pi_{P} \left( x \right) = \sqrt {1 - \left( {\mu_{P} \left( x \right)} \right)^{2} - \left( {v_{P} \left( x \right)} \right)^{2} }\), \(\left( {\mu_{P} \left( x \right), \, v_{P} \left( x \right)} \right)\), \(\alpha = \left( {\mu_{\alpha } ,v_{\alpha } } \right)\), $$ s\left( \alpha \right) = \mu_{\alpha }^{2} - v_{\alpha }^{2} ,\quad {\text{where}}\quad s\left( \alpha \right) \in \left[ { - 1,1} \right]. Based on the \(w_{{j_{0} j}}\) obtained in Sect. $$, \(\alpha^{ + } = \left( {\max_{j} \left\{ {\mu_{j} } \right\},\min_{j} \left\{ {v_{j} } \right\}} \right),\alpha^{ - } = \left( {\min_{j} \left\{ {\mu_{j} } \right\},\max_{j} \left\{ {v_{j} } \right\}} \right)\), $$ \min_{j} \left\{ {\mu_{j} } \right\} \le \mu_{i} \le \max_{j} \left\{ {\mu_{j} } \right\},\max_{j} \left\{ {v_{j} } \right\} \ge v_{i} \ge \min_{j} \left\{ {v_{j} } \right\}. 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. The ranking results by using the PFWIBM operator are different when the parameters \(p = {1},q = {40}\) and \(p = {1},q = {50}\). People in business often calculate weighted mean to evaluate the average prices of goods purchased from different vendors where the quantity is considered the weight. For example, a student may use a weighted mean in order to calculate his/her percentage grade in a course. Based on the comprehensive ratings \(sc_{i}\) to get the ranking of alternatives \(a_{\sigma \left( i \right)} \succ a_{{\sigma \left( {i + 1} \right)}}\). Math. The stock returns could get affected. J. $$, $$ \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} . Im sure there are more efficient ways, but one solution might be to use a for-loop. 608, 153177 (2022), Banerjee, D., Dutta, B., Guha, D., et al. Obviously, the INWIBM operator still satisfies the properties of Theorem 1. To prove the PFWIBM operator is also a PFN i.e., Because the PFN \(\alpha_{i} = \left( {\mu_{i} ,v_{i} } \right)\left( {i = 1,2, \cdot \cdot \cdot ,n} \right)\) satisfies \(\mu_{i} \le N_{d} \left( {v_{i} } \right)\), according to the monotonicity of \({\text{INWIBM}}^{p,q}\), we have. 183, 115368 (2021), Article Mean is an average of given numbers. Step 3. $$, \(w_{{j_{0} j}} \left( {j_{0} \ne j;j_{0,} j = 1,2, \ldots ,n} \right)\), $$ w_{{j_{0} j}} = \varepsilon ru_{{j_{0} j}} + \left( {1 - \varepsilon } \right)re_{{j_{0} j}} ,\varepsilon \in \left[ {0,1} \right]. 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. \right\}\), \(\gamma \left( {\widetilde{cu}_{{i,gj_{{0}} }}^{{}} ,\widetilde{cu}_{i,gj}^{{}} } \right)\), \(\widetilde{cu}_{{i,gj_{{0}} }}^{{}} \in \widetilde{cu}_{{i,gj_{{0}} }}^{{}}\), \(\widetilde{cu}_{i,gj}^{{}} \in \widetilde{cu}_{i,gj}^{{}}\), $$ \gamma \left( {\widetilde{cu}_{{i,gj_{{0}} }}^{{}} ,\widetilde{cu}_{i,gj}^{{}} } \right) = \frac{{\mathop {\min }\limits_{i} \mathop {\min }\limits_{{j \in \left\{ {{\text{Index}}\left( {c_{j} } \right)\left| {c_{j} \in I_{{j_{0} }} } \right.} Ans. The DEMATEL method is an effective method for factor analysis and identification through expert judgment [30]. It is the aggregate of all the values in a data set divided by the total count of the observations. In case you want to learn more the theoretical research concept of the weighted arithmetic mean, I can recommend the following video tutorial of the MySecretMathTutor YouTube channel: In addition, you may also have a look at some of the related R tutorials that I have published on this website: I hope this tutorial contained the content you were looking for. Appl. Int. 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. According to the criteria weights \(w_{{j_{0} }}^{c}\) and the comprehensive interaction coefficients \(w_{{j_{0} j}}\), the Pythagorean fuzzy matrix \(\widetilde{CU}\) is aggregated by Eqs. data <- data.frame(x1, w1, group) # Create data frame. 72201097), the Ministry of Education of Humanities and Social Science Foundation of China (Grant no. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. An intuitionistic fuzzy set (IFS) \(I\) in \(X\) is given by. $$, \(\alpha_{i} = \beta_{i} = \left( {\mu_{{\beta_{i} }} ,v_{{\beta_{i} }} } \right),i = 1,2, \ldots ,n\), $$ {\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). Fuzzy Syst. \right\}}} d\left( {\widetilde{cu}_{{i,gj_{{0}} }}^{{}} ,\widetilde{cu}_{i,gj}^{{}} } \right) + \rho \mathop {\max }\limits_{i} \mathop {\max }\limits_{{j \in \left\{ {{\text{Index}}\left( {c_{j} } \right)\left| {c_{j} \in I_{{j_{0} }} } \right.} If \(w_{1,2}^{\left( 1 \right)} > w_{1,2}^{\left( 2 \right)}\), by Property 2, we obtain \(\Delta_{{x_{1} ,x_{2} }}^{\left( 1 \right)} > \Delta_{{x_{1} ,x_{2} }}^{\left( 2 \right)}\), then \(\Delta_{{x_{1} }}^{\left( 1 \right)} > \Delta_{{x_{1} }}^{\left( 2 \right)}\) and \(\Theta_{{x_{1} }}^{\left( 1 \right)} > \Theta_{{x_{1} }}^{\left( 2 \right)}\). The mutual coefficient structure embedded in NWIBM operator lays a good foundation for the operator proposed in this paper, and provided a better solution for quantifying the degree of attribute interaction. It sums up the numbers and divides them with the count of numbers which provides us with the mean. (24) and shown in Table 9, and then the criteria weights \(w_{{j_{0} }}^{c} = \left( {0.12,0.12,0.13,0.12,0.13,0.12,0.11,0.15} \right)\) are obtained through Eqs. [27] Let \(\alpha = \left( {\mu_{\alpha } ,v_{\alpha } } \right)\) and \(\beta = \left( {\mu_{\beta } ,v_{\beta } } \right)\) be two PFNs, the distance between \(\alpha\) and \(\beta\) is defined as. Comput. By combining INWIBM and its dual operator, the Pythagorean fuzzy Bonferroni mean with weighted interaction (PFWIBM) operator is proposed. 11(1), 10911110 (2018), Akram, M., Wasim, F., Alcantud, J.C.R., et al. In addition, because Pythagorean fuzzy sets contain intuitionistic fuzzy sets, that is, any IFSs are PFSs. $$, $$ \begin{gathered} {\text{PFWIBM}}^{p,q} \left( {\alpha^{ - } ,\alpha^{ - } , \ldots ,\alpha^{ - } } \right) \le {\text{PFWIBM}}^{p,q} \left( {\alpha_{1} ,\alpha_{2} , \ldots ,\alpha_{n} } \right), \hfill \\ {\text{PFWIBM}}^{p,q} \left( {\alpha_{1} ,\alpha_{2} , \ldots ,\alpha_{n} } \right) \le {\text{PFWIBM}}^{p,q} \left( {\alpha^{ + } ,\alpha^{ + } , \ldots ,\alpha^{ + } } \right). : Application of Choquet integral in interval type-2 Pythagorean fuzzy sustainable supply chain management under risk. \kern-\nulldelimiterspace} 2}}} . In contrast, the operators proposed in this paper can better handle such situations. weighted_columns[i] <- weighted.mean(data[ , i], w1) According to Table 15, it can be concluded that the total ratings of the cars increases with increasing values of \(p\) and \(q\). 500, 229258 (2019). : Multi-criteria optimization technique with complex pythagorean fuzzy n-soft information. https://doi.org/10.1109/TFUZZ.2022.3179594, Demir, U.Y., Diner, H., Yksel, S., et al. The grey relational coefficient between the criterion \(c_{{j_{0} }}\) and its associated set of criteria \(c_{j}^{i} \in I_{{j_{0} }}\) can be calculated as follows, where \(j \in \left\{ {{\text{Index}}\left( {c_{j} } \right)\left| {c_{j} \in I_{{j_{0} }} } \right.} \[\frac{\sum_{i=1}^{4}w_{i}x_{i}}{\sum_{i=1}^{4}w_{i}}\] = \[\frac{2566}{1000}\], The weighted arithmetic mean of mobiles per household in this sample is 2.566. Different BM operators with \(p = q = 1\) are used to obtain the aggregation results, as shown in Table 13. J. Pythagorean Fuzzy Bonferroni Mean with Weighted Interaction Operator and Its Application in Fusion of Online Multidimensional Ratings. Step 1. Step 4. 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. Data values with high weights contribute to the more weighted mean than the weights with lower weighted mean. (Continuation of Example 1) Let \(w_{1,2}^{\left( 1 \right)} > w_{1,2}^{\left( 2 \right)}\), by Property 2, we have \(\Delta_{{x_{1} ,x_{2} }}^{\left( 1 \right)} > \Delta_{{x_{1} ,x_{2} }}^{\left( 2 \right)}\), then \(\Delta_{{x_{1} }}^{\left( 1 \right)} > \Delta_{{x_{1} }}^{\left( 2 \right)} ,\Theta_{{x_{1} }}^{\left( 2 \right)} > \Theta_{{x_{1} }}^{\left( 1 \right)}\). Weighted mean (also known as weighted average) is the average of the given data set. This number of 42% is much closer to equity returns of 50%, as equity accounts for most of the portfolio. J. Appl. Table 1: Example Data with Numeric Column, Weights & Group Indicator. Syst. 4.1, an online multi-dimensional rating aggregation decision-making approach is developed. Int. The weighted arithmetic mean will be equal to an ordinary arithmetic mean in case all the given weights are equal. The following are 5 numbers and the weights assigned to each number. In Sect. Lets start with the weightedMean function of the matrixStats package. 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. Now, we can calculate the weighted mean with the following R code: data %>% # Weighted mean by group
: A novel VIKOR approach based on entropy and divergence measures of Pythagorean fuzzy sets to evaluate renewable energy technologies in India. $$, $$ a_{\sigma \left( i \right)} \succ a_{{\sigma \left( {i + 1} \right)}} ,{\text{where}}\quad sc_{\sigma \left( i \right)} > sc_{{\sigma \left( {i + 1} \right)}} . Then, calculate the comprehensive interaction coefficients \(w_{{j_{0} j}}\) based on \(r\_expert\) and \(r\_user\). A Pythagorean fuzzy set (PFS) \(P\) in \(X\) is given by. w = corresponding weight for each observation. Some weights maybe 0 but not all as the division by 0 is not permitted. YF performed the validation and formal analysis. 584, 6588 (2022), Liu, P., Rani, P., Mishra, A.R. 569, Yuelu Avenue, Changsha, 410205, China, Yi Yang,Feifan Yang,Jie Chen,Yangyan Zeng&Limei Liu, School of Digital Media and Humanities, Hunan University of Technology and Business, No. To compute the weighted mean by group we can use the functions of the dplyr package. This article contains five examples including reproducible R codes. Find the weighted mean for the given data set. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Weighted means are useful in a wide variety of scenarios. Thus, if one performs poorly in chapter tests but does well in final exams, the weighted average of the grades will be relatively high. f and m)? A weighted average or weighted mean is used when some data values are considered to be more important than other values and so we want them to contribute more to the final 'average'. (23), the set of criteria interaction \(I_{j}\) is obtained, as shown in Table 8, where the \(\theta = 0.6815\). Inf. For instance, let us assume equity consists of 80% of a portfolio and debt balance 20%. Finally, the proposed method is applied to a passenger car ranking example to show the feasibility of the method, and through some comparative analysis, the advantages of the proposed operator are clarified, and the influence of operator parameters on the decision-making results is analyzed. After installing and loading the R package, we can apply the wt.mean command as follows: wt.mean(x1, w1) # Apply wt.mean function
512, 14811502 (2020), Fernndez, E., Navarro, J., Solares, E.: A hierarchical interval outranking approach with interacting criteria. Accordingly, the improved NWIBM operator is defined as follows. [6] Let \(X\) be a universe of discourse. library("SDMTools") # Load SDMTools package. Int. Presentation in tabular form is not compulsory but makes the calculations easy. How to find weighted standard deviation or mean for many columns in the database. $$, $$ sc_{i} = \sum\limits_{g = 1}^{{u_{i} }} {\widetilde{d}_{g}^{i} /u_{i} } . : A novel Pythagorean fuzzy combined compromise solution framework for the assessment of medical waste treatment technology. $$, \(\beta = \left( {\mu_{\beta } ,v_{\beta } } \right)\), $$ d\left( {\alpha ,\beta } \right) = \frac{1}{2}\left( {\left| {\left( {\mu_{\alpha } } \right)^{2} - \left( {\mu_{\beta } } \right)^{2} } \right| + \left| {\left( {v_{\alpha } } \right)^{2} - \left( {v_{\beta } } \right)^{2} } \right| + \left| {\left( {\pi_{\alpha } } \right)^{2} - \left( {\pi_{\beta } } \right)^{2} } \right|} \right). The negative weights are not possible, some may be zero but not all of them as division by 0 is not allowed. Correspondence to Inf. - 51.91.151.248. IEEE Trans. 357, 6187 (2016), Article Expert Syst. This basically occurs similarly as some professors or teachers assign grades for their courses. 35(3), 401431 (2020), Liang, Y., Qin, J., Martnez, L., et al. Solution: Step 1: To get the sum of weighted terms, multiply each average by the number of students that had that average and then sum them up. $$, \(w_{i} = {1 \mathord{\left/ {\vphantom {1 n}} \right. Frequent interactions between attributes in real decision-making problems [13,14,15], such as the correlation between the interior decoration and comfort of automobile products, promoted the interactive decision-making of criteria to become one of the important branches of multi-criteria decision-making [13,14,15]. If you accept this notice, your choice will be saved and the page will refresh. Intell. Multiply each number and the relevant weight assigned to that number (w1 by x1, w2 by x2, and so on). In this section, the proposed approach is used to ranking passenger cars from autohome.com. (2022). (3133) to obtain the user comprehensive rating matrix \(\widetilde{D}\), where \(p = q = 1\). document.getElementById( "ak_js_1" ).setAttribute( "value", ( new Date() ).getTime() ); Im Joachim Schork. It can be concluded that \(w_{i,j}\) is inversely proportional to \(\Theta_{{x_{i} }}\). The sum of rows and the sum of columns are separately denoted as D and R within the comprehensive impact matrix \(M\). Based on \(\overline{\gamma }\left( {c_{{j_{0} }} ,c_{j} } \right)\), the user criterion interaction coefficient matrix \(r\_user\) is obtained. However, these existing operators rely heavily on the subjective settings of the decision-maker to determine the interaction or independence between criteria. Check out the video below where I explain the steps of this article more detailed: Please accept YouTube cookies to play this video. Part of Springer Nature. MathSciNet In the following two examples, I will therefore show some alternatives to the weighted.mean function. to one organization and as a liability to another organization and are solely taken into use for trading purposes.read more. Prod. Fuzzy Syst. B 41(2), 568578 (2011), Chen, Z.S., Yang, Y., Wang, X.J., et al. The weighted mean equation is a statistical method that calculates the average by multiplying the weights with their respective mean and taking its sum. \\ \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} . Obtain the criteria interaction matrix \(r\_expert\) provided by the expert. You are free to use this image on your website, templates, etc., Please provide us with an attribution link. 23(1), 251267 (2019), Chen, Z.S., Chin, K.S., Tsui, K.L. Syst. In this paper, we address the problem of multi-criteria of products and uncertainty of user ratings to effectively aggregate online ratings to facilitate information screening by consumer groups. 3.1, the dual of the INWIBM operator is defined. \right\} = \left\{ {\left( {1,0} \right),\left( {0.8,0.3} \right),\left( {0.6,0.5} \right),\left( {0.3,0.9} \right),\left( {0,1} \right)} \right\}\), \(\left| {sc_{2} - sc_{3} } \right| < 0.1\), https://doi.org/10.1007/s44196-022-00152-y, https://doi.org/10.1109/TFUZZ.2022.3179594, http://creativecommons.org/licenses/by/4.0/. The current study is arranged as the following. Calculate criterion weights \(w_{{j_{0} }}^{c}\)[33]. To explore the advancement of the operator, the criteria interaction coefficients of BM operators in the literature [19, 21, 25, 37] is analyzed. To solve the problem in Sect. A weighted average, on the other hand, is an average calculated in the same way but with a weight multiplied with each data set. Ans. Int. Step 4. For instance, one generally uses it to calculate a specific courses final grade. \\ \end{aligned} $$, \(\alpha_{i} = \left( {\mu_{i} ,v_{i} } \right)\left( {i = 1,2, \cdot \cdot \cdot ,n} \right)\), \(\mu_{i} \le N_{d} \left( {v_{i} } \right)\), $$ \begin{aligned} & N_{d} \left( {{\text{INWIBM}}_{{{\text{dual}}}}^{{^{p,q} }} \left( {v_{1} ,v_{2} , \ldots ,v_{n} } \right)} \right) \\ & \quad {\text{ = INWIBM}}^{p,q} \left( {N_{d} \left( {v_{1} } \right),N_{d} \left( {v_{2} } \right), \ldots ,N_{d} \left( {v_{n} } \right)} \right) \ge {\text{INWIBM}}^{p,q} \left( {\mu_{1} ,\mu_{2} , \ldots \mu_{n} } \right). Select a blank cell and type this formula = SUMPRODUCTSUMPRODUCTThe SUMPRODUCT excel function multiplies the numbers of two or more arrays and sums up the resulting products.read more(B2: B5, C2: C5), where the range B2: B5 represents the weights and the range C2: C5 represents the numbers. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. In this study, the operation mechanism of interaction coefficients with the support of a large number of actual ratings is optimized, and a two-stage interaction coefficient calculation method that combines expert knowledge with actual ratings is proposed. Syst. In Sect. It is an integral part of the discounted valuation analysis which calculates the present value of a firm by discounting future cash flows by the expected rate of return to its equity and debt holders.read more is 6%. Step 2: Calculate the numerator of the weighted mean formula, Multiply each grade by its counts and then add the product together, \[\sum_{i=1}^{4}\]w\[_{i}\]x\[_{i}\] = w\[_{1}\]x\[_{1}\] + w\[_{2}\]x\[_{2}\] + w\[_{3}\]x\[_{3}\], = (1)(73) + (2)(378) + (3)(459) + (4)(90). $$, $$ {\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).