Weighted Aggregated Sum Product Assessment

Multiple Criteria Decision-Making (MCDM) techniques are widely employed in industry and academia. The Weighted Product Model (WPM) and the Weighted Sum Model (WSM) are two of the most common techniques among all MCDMs [1-6]. However, when WASPAS (weighted aggregated sum product assessment) decision-making technique first was suggested in 2012, it was considered to be among the most compelling alternative MCDM techniques because it combines the Weighted Product Model (WPM) and the Weighted Sum Model (WSM) into a single procedure [5, 7].

The WASPAS technique was offered by Zavadskas et al. [6]; and it in successful subset of the recent generation of MCDMs [5, 6, 8]. It is argued that WASPAS' algorithm is simple, and it is capable of producing more accurate decision outcome than classic WSM and WPM techniques [5, 7, 9]. WASPAS has received substantial attention from decision makers from all sorts of backgrounds due to the simplicity of the computing process and the accuracy of the outputs, and it is now widely referenced as an excellent decision support tool [5, 7].

Based on previous practices [10-13], WASPAS's utilization necessitates four following steps for analyses of the data:

(1) Creation of a decision matrix, X=[x(i,j)]_mn, where n is the number of criteria, m is the number of alternatives, and x_ij is the performance of the ith alternative in relation to the jth criterion.

(2) Normalization of all entries in the decision matrix using the following two equations to make the metrics non-dimensional. The formula, where the normalized value of x(i,j) is x^´(i,j), for beneficial criteria is x^´(i,j)=x(i,j)/maximum_i x(i,j) and for non-beneficial criteria it is x^´(i,j)=minimum_i x(i,j) /x(i,j).

(3) The first relative significance of the ith option, in one hand, is analogous to the WSM technique. The Q⁽¹⁾i= $\sum_{j=1}^n X(\mathrm{i}, \mathrm{j}) \cdot W(\mathrm{j})$  formula is used to compute the overall relative significance of the ith alternative where w(j) is the relative significance (weight) of the jth criteria. The second relative significance of the same ith alternative, in the other hand, is calculated using the WPM approach using the Q⁽²⁾_i $\prod_{j=1}^n X(\mathrm{i}, \mathrm{j})^{\wedge} W(\mathrm{j})$  formula.

(4) Q_i=λQ⁽¹⁾_i+(1 - λ)Q⁽²⁾_i is a generalized equation established in WASPAS for calculating the total relative significance of ith alternative, where λ is the combination parameter in the range of 0 to 1. WASPAS method is transformed into WPM when the value of λ is 0, and WSM method when it is 1. λ is been used to remedy MCDM issues in ranking accuracy.

A case is borrowed to practice the WASPAS MCDM, referencing an application of WASPAS from a reference [14] and simplified here. In a decision, there are 3 criteria (C1, C2 and C3), the first two of which is beneficial but the second one is cost-related (not beneficial). C1 and C2 are equal important, 25%, but C3 is more important, 50%. Based on MCDM WASPAS, the decision maker has 5 alternatives from which to choose. Their decision matrix in shown via Table 1, as step 1 of WASPAS. Normalization of all entries in the decision matrix in shown in Table 2, as WASPAS step 2.

Multiplying the relative importance of each criterion leads us to the next step of attribute optimization. For one, the first cell followed by WSM approach gives A1 as Q⁽¹⁾_i: = $\sum_{j=1}^n X(\mathrm{i}, \mathrm{j}) \cdot W(\mathrm{j})$=(5/5×0.25)+(3/5×0.25)+(8000/8500×0. 50). However WPM approach for A1 gives Q⁽²⁾_i =$\prod_{j=1}^n X(\mathrm{i}, \mathrm{j})^{\wedge} W(\mathrm{j})$= (5/5$\wedge$0.25) × (3/5$\wedge$0.25) × (8000/8500$\wedge$0. 50). Finally, the last step in WASPAS, after getting Q⁽¹⁾_i and Q⁽²⁾_i for all alternatives, is finding Q_i=λQ⁽¹⁾_i+(1- λ)Q⁽²⁾_i as ,with decided λ =1/2 , it became (Q⁽¹⁾_i/2)+(Q⁽²⁾_i /2) for all alternatives. WASPAS results in choosing the best alternatives as shown in Table 3.

Table 1. Decision matrix

Table 2. Normalized matrix

Table 3. Ranking of alternatives

WASPAS method has been used in various research projects, however there is no synopsis of its usage yet. This study attempts to fill the literature gap by conducting a WASPAS literature survey.

As illustrated in Figure 1, the use of the WASPAS method in scientific documents is steadily expanding year after year, according to Scopus statistics. Among the WASPAS users, Edmundas Kazimieras Zavadskas from Vilnius Gediminas Technical University, who developed the method, was strongly top author listed based on the archived document number, with 11.4 percent of the documents. His university colleague, Jurgita Antucheviciene, was second listed with 3.9 percent of the documents, and another university colleague, Zenonas Turskis, was next with 3.3 percent of the documents.

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Figure 1. Documents by year

Using the data of Figure 1, results Figure 2 with a non-linear regression equation based on the Power Model with N publications and A years of method’s age as A=0.03N^3.53; The response variable in a power mode non-linear regression is related to the factor increased to a power [20]. That outcome predicts, for instance, about 400 published documents are likelihood to be archived via Scopus in 2025. Plotting visualization of Figure 2 provides the fitting validation.

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Figure 2. Publications prediction

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Figure 3. Overlay visualization of co-authorships

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Figure 4. Network visualization of Keywords

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Figure 5. Keywords Evolution over time

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Figure 6. Authors, keywords, and journals

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Figure 7. Documents by country