Improvement Ranking Accuracy of Weighted Aggregated Sum Product Assessment With Lambda Variable

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DOI: 10.30865/mib.v7i1.5280

Improvement Ranking Accuracy of Weighted Aggregated Sum Product Assessment With Lambda Variable

Muhadi M. Ilyas Gultom, Erna Budhiarti Nababan^*, Zakarias Situmorang Informatics Engineering, University of North Sumatra, Medan, Indonesia

E-mail: ¹[email protected], ^2,*[email protected], ³[email protected] Correspondence Author Email: [email protected]

Abstract−Conventional methods are still used in selecting the best students in the various institutions depending on the subjectivity of each member of the assigned committee. In order to make an objective decision, it is necessary to have a method that can consider the criteria used to select the candidates to be elected. The decision-making method used in this study is Weighted Aggregated Sum Product Assessment(WASPAS). This study aims to analyze the increase in accuracy of the WASPAS method that occurs in the implementation of the lambda variable in the process of combining the Weight Product Method(WPM) and Weight Sum Method(WSM). This method is use because it is suitable for the case studied where the application of this method focuses on weighting criteria with a dynamic number of alternatives and low computational complexity providing good performance in handling large amounts of data.The application of this method uses data from students from Engineering Faculty of Universitas Islam Sumatera Utara which is tested on 10 students with criteria adapted from student data attributes that can be used as parameters for decision making. The results of this study show an increase for each alternative with an average value of 23.6% for each alternative. From this study it can be concluded that accuracy is highly dependent on variations in lambda values which are affected by the determinant operator in the equation used. Therefore it is possible to find an absolute equation to give optimal effect on a single value without variation by considering the bias of the effect of the WASPAS method on the lambda variable in future research.

Keywords: MCDM; WASPAS; Variable Lambda

1. INTRODUCTION

Weighted aggregated sum product assessment is a method that is very suitable for multi-criteria decision making with many and dynamic alternatives. This method has advantages in terms of accuracy and also clear logic and a simple calculation process [1]. This method is a combination of the weighted sum method and weighted product method that produce a better level of accuracy than the initial method [2]. The WASPAS method has a high level of practice and strongly refers to the concept of ranking accuracy [3]. Most published research ignores the concept of ranking accuracy and the WASPAS method is usually devoted to that problem [4]. Although many applications have implemented the waspas method successfully.

In several studies conducted it has been proven that the alert method has a high level of accuracy and consistency [5] in making decisions, where the test is carried out by making a comparative analysis [6] of popular methods that are on the topic of decision making with multiple criteria. From this comparison it can be seen clearly how the multi-criteria decision-making methods provide a logical assessment of the cases handled and it is clear that the WASPAS method has low reliability and low computational complexity so that it is easy to observe and detect illogical decision gaps in the results.

Selection of the best students is a way to assess students who are eligible to receive various predicate such as awards and scholarships for further education. [7]. The selection of the best students in various institutions still uses conventional methods that depend on the subjectivity of each individual from the members of the committee.To avoid subjectivity which is considered as human error which is often influenced by the environment and things that do not have significant weight for wise decision making. In order to make an objective decision, a method is needed that will consider the criteria that will be used to select the candidate.

In this study, the selection of the best students was applied to the WASPAS method normally and compared to the WASPAS method which uses lambda variables in combining the values of the WSM and WPM methods to improve the ranking accuracy of each alternative. The data used in this study were sourced from the engineering faculty of the Islamic University of North Sumatra. In determining the criteria, the researcher selects data attributes that have a range of values in the form of qualitative variables which are converted into quantitative data as variables that will be used in calculations using the WASPAS method. In this study, the Waspas method was first equipped with a level value determined by the researcher because this method did not yet have an official level of importance in the original model [8]. Therefore, this value is usually determined by researchers or obtained from values from scales that are widely used as research instruments such as the Likert scale or the scale value from the fuzzy method [9] either by crisp method, using linguistic variables or other ways used for the development of a decision-making method with many criteria [10].

Based on the application, the WASPAS method is commonly used in selecting various research objects.

Such as selecting the right material handling equipment on conveyors and automated guided vehicles (AGVs) [11], selecting real time industrial robots [12], path selection for safe evacuation routes [13], selecting appropriate alternative-fuel for vehicles [14], selecting healthcare waste disposal location [15] and many more. Several trials in an effort to increase accuracy in the decision making method are also carried out, such as the use of fuzzy in the

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This study uses weighting parameters determined by the researchers themselves based on considerations from previous studies which have differences in the percentage of weights for each quality parameter used. One of the advantages that led to the choice of this method is its low computational complexity, thus providing better computational performance compared to other methods for dynamic and increasing amounts of data.

This study uses a case study of selecting the best students at the Faculty of Engineering at the Islamic University of North Sumatra in the analysis of increasing accuracy in the WASPAS method using the lambda variable. The aspects used in selecting the best students in this study were academic grades, economic conditions, semester, and student youth. All the variables used were obtained from observing previous studies [18] with similar problems [19] which were then adjusted to the data used in this study.

2. RESEARCH METHODOLOGY

This research phase is a comparative analysis on ranking accuracy using the normal WASPAS method and the WASPAS method using the lambda variable to select the best student. The application of the method using the data that has been collected and the determination of criteria as well as sub-criteria are determined based on predetermined aspects.

2.1 System Design

In this study, the system was designed by initially collecting data from student data taken from the Islamic University of North Sumatra. Then the data that has been collected is selected to obtain data that has complete attributes that are used for this research experiment. After the data selection is complete, the criteria to be used as well as the range of sub-criteria are determined. After that, proceed to the initial decision matrix development stage for each alternative based on predetermined criteria. The complete decision matrix is then normalized by first determining the highest and lowest values. This normalization is done to prevent errors in calculations caused by large differences in values. After the decision matrix is normal, the WASPAS method calculation begins where the calculation is carried out by constructing the WSM (weighted sum method) matrix and WPM (weighted product method) matrix.

In the general WASPAS method, after the two matrices have been built, calculations are carried out to obtain the total joint, followed by ranking all the alternatives tested. However, for the WASPAS method, which wants to improve its accuracy, a search for the value of the lambda variable is first carried out from the results of calculations using the results from the WSM and WPM matrices. After that, for the merging process, the optimal value of the variation in element values is determined in advance with a range of 0-1. Then after these two values are obtained, a calculation is carried out to find the total joint which ends with ranking all alternatives. The alternative with the highest total joint value is the alternative with the highest rank. The flowchart of the system design that has been described is illustrated in figure 1.

Figure 1. System Design 2.2 Data Source

The data that has been collected is sourced from student data from the Faculty of Engineering, Islamic University of North Sumatra. The raw data obtained was 1053 student data where the data was re-selected based on the

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DOI: 10.30865/mib.v7i1.5280

Muhadi M. Ilyas Gultom, Copyright © 2023, MIB, Page 565 completeness of the attributes required by the criteria used. After going through the selection, 70 student data were obtained that were feasible to be tested as test material. In this study, the top 10 sample data were taken alphabetically from the appropriate student data that shown in table 1.

Table 1. Table of data source

Code Students Criteria

Academic Score Age Parent's Income Semester

A1 Arief Sofiansyah 3.78 20 Rp.1,000,000 - Rp.1,999,999 2

A2 Aktha Mathias H. 3.72 33 Rp.2,000,000 - Rp.4,999,999 2

A3 Aldo Romadhon 3.53 22 Rp. 500,000 - Rp. 999,999 2

A4 Ayatullah Komaini 3.76 33 Rp.5,000,000 - Rp. 20,000,000 2

A5 Dedek Pratama 2.48 21 Rp.1,000,000 - Rp.1,999,999 2

A6 Dimas Ardi Pratama 3.71 19 Rp.1,000,000 - Rp.1,999,999 2

A7 Dimas Pradipta 3.03 22 Rp.5,000,000 - Rp. 20,000,000 4

A8 Hervan Fernando S. 3.02 24 Rp.2,000,000 - Rp.4,999,999 4

A9 Igo Purnama 3.66 19 Kurang dari Rp. 500,000 4

A10 Lalu Azani S. Azmi 3.73 19 Rp.2,000,000 - Rp.4,999,999 2 2.3 Determining Criteria

Determination of criteria is done by adapting attribute data to the data obtained with the specified aspects. The following are the criteria determined by the researchers based on aspects of academic value, economic conditions and student youth. The determination of the weight on each criterion is determined from the quality of the criteria based on the type of criteria that are advantages or disadvantages. The criteria used were obtained from considering the data attributes in this research case and also by highlighting the effect of each criterion on general cases. The weight value of each of the following criteria is obtained from the total percentage of all criteria distributed based on the influence of these criteria that determined by the researcher shown in table 2.

Table 2. Table of criteria

Code Name of Criteria Type Weight C1 Academic Score Benefit 0.40

C2 Age Cost 0.15

C3 Parent’s Income Cost 0.30

C4 Semester Cost 0.15

2.4 Determining Sub-Criteria

Sub-criteria is a range of qualitative data with a level of importance that will be weighted to be a quantitative variable as a calculation material in this study. Sub-criteria is made to classify the weights on each of the existing criteria. The following is a table that describes the level of importance and its weight. the determination of the value of each sub-criteria is obtained from a method that is often used as a research instrument, namely the Likert scaling value. After being assessed, it is converted into a quantitative value using the Saaty scaling where this scaling gives a fixed weight at each existing level. These values can be described in terms of importance as in table 3 below.

Table 3. Table level of importance Weight Level

1 Worst

2 Bad

3 Average

4 Good

5 Best

Based on the level of importance above, the data range is weighted for each of the existing criteria.Table 4 below shows the results of the sub-criteria weighting for each criterion.

Table 4. Table of subcriteria

Criteria Code Data Range Weight

C1

< 2 1

> 2 and <= 2.5 2

> 2.5 and < 3 3

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Criteria Code Data Range Weight

> 3 and < 3.5 4

> 3.5 5

C2

<18 1

18 2

19 3

20 4

>20 5

C3

< Rp. 500,000 1 Rp. 500,000 - Rp. 999,999 2 Rp.1,000,000 - Rp.1,999,999 3 Rp.2,000,000 - Rp.4,999,999 4 Rp.5,000,000 - Rp. 20,000,000 5

C4

2 1

4 2

6 3

8 4

> 8 5

From the table above, it can be seen that the weight range for each criterion is determined based on the level of importance made before. where the code criteria represent each criterion used, the data range represents the range of qualitative data from student data attributes and the weights represent the level of importance.

2.4 Developing Initial Decision Matrix

By determining the weight values of the criteria and sub-criteria, a decision matrix can be formed which is the basis for applying the WASPAS method calculations. At this stage each alternative will be weighted based on the importance of each criterion used. The initial matrix is made based on student data where each data attribute has been given a weight according to the specified sub-criteria value. Each multi-criteria decision-making problem starts with the following matrix :

𝐴 = [

𝐴

₁₁

𝐴

₁₂

𝐴

₁₃

𝐴

_1𝑗

𝐴

₂₁

𝐴

₂₂

𝐴

₂₃

𝐴

_2𝑗

𝐴

₃₁

𝐴

₃₂

𝐴

₃₃

𝐴

_3𝑗

⋮ ⋮ ⋮ ⋮ 𝐴

_𝑖1

𝐴

₁₂

𝐴

₁₃

𝐴

_𝑖𝑗

]

Where i is the number of students and j is the number of criteria that have been evaluated based on their respective weights. The result of the initial decision making matrix shown in table 5.

Table 5. Table of initial decision matrix Student

Code

Criteria Code

C1 C2 C3 C4

A1 5 4 3 1

A2 5 5 4 1

A3 5 5 2 1

A4 5 5 5 1

A5 2 5 3 1

A6 5 3 3 1

A7 4 5 5 2

A8 4 5 4 2

A9 5 3 1 2

A10 5 3 4 1

The table above explains the value of the weight of the sub-criteria for each student represented by the student code on each criterion represented by the criteria code.

2.5 Developing Normalization Decision Matrix

In multi-criteria decision making, each criteria has 2 different types including beneficial criteria and cost criteria.

For each type of criteria must be normalized linearly on each element of the matrix using the following equation.

Equation for criteria with benefit type use the equation 1 :

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DOI: 10.30865/mib.v7i1.5280

̅̅̅̅ = ^{min A}^ij

Aij (1)

Where Aij is the weight of each student with respect to the criteria used and min Aij is the lowest score on i row of the student weight in the j criteria column.

Equation for criteria with cost type use the equation 2 : 𝐴_𝑖𝑗

̅̅̅̅ = ^𝐴^𝑖𝑗

𝑚𝑎𝑥 𝐴_𝑖𝑗 (2)

Where Aij is the weight of each student with respect to the criteria used and max Aij is the highest score on i row of the student weight in the j criteria column. Āij the accent on the symbol indicates that the candidate weights have been normalized. The type of each criterion can be seen in table 2. In this case, the matrix is described as follows:

𝐴 =

[ 𝐴₁₁

𝑚𝑎𝑥 𝐴_𝑖1 𝑚𝑖𝑛 𝐴_𝑖2

𝐴₁₂ 𝑚𝑖𝑛 𝐴_𝑖3

𝐴₁₃ 𝑚𝑖𝑛 𝐴_𝑖𝑗 𝐴_1𝑗 𝐴₂₁

𝑚𝑎𝑥 𝐴_𝑖1

𝑚𝑖𝑛 𝐴_𝑖2

𝐴₂₂ 𝑚𝑖𝑛 𝐴_𝑖3

𝐴₂₃ 𝑚𝑖𝑛 𝐴_𝑖𝑗 𝐴_2𝑗 𝐴₃₁

𝑚𝑎𝑥 𝐴_𝑖1 𝑚𝑖𝑛 𝐴_𝑖2

𝐴₃₂ 𝑚𝑖𝑛 𝐴_𝑖3

𝐴₃₃ 𝑚𝑖𝑛 𝐴_𝑖𝑗 𝐴_3𝑗

⋮ ⋮ ⋮ ⋮ 𝐴_𝑖𝑗

𝑚𝑎𝑥 𝐴_𝑖𝑗

𝑚𝑖𝑛 𝐴_𝑖2

𝐴_𝑖2 𝑚𝑖𝑛 𝐴_𝑖3

𝐴_𝑖3 𝑚𝑖𝑛 𝐴_𝑖𝑗 𝐴_𝑖𝑗 ]

Based on the matrix above, the calculation can be done as below :

A1 : A11 = (5/5), A12 = (4/3), A13 = (3/1), A14 = (1/1)

A11 = 1, A12 = 0.75, A13 = 0.33, A14 = 1

A2 : A21 = (5/5), A22 = (5/3), A23 = (4/1), A24 = (1/1)

A21 = 1, A22 = 0.60, A23 = 0.25, A24 = 1

A3 : A31 = (5/5), A32 = (5/3), A33 = (2/1), A34 = (1/1)

A31 = 1, A32 = 0.60, A33 = 0.50, A34 = 1

⫶ ⫶ ⫶ ⫶ ⫶

A10 : A10,1 = (5/5), A10,2 = (3/3), A10,3 = (1/1), A10,4 = (2/1)

A10,1 = 1, A10,2 = 1, A10,3 = 0.25, A10,4 = 1

After performing calculations on each element of the matrix using the above equation, the matrix value is obtained which is shown in table 6.

Table 6. Table of normalization decision matrix Student

Code

Criteria Code

C1 C2 C3 C4

A1 1 0.75 0.33 1

A2 1 0.60 0.25 1

A3 1 0.60 0.50 1

A4 1 0.60 0.20 1

A5 0.4 0.60 0.33 1

A6 1 1 0.33 1

A7 0.8 0.60 0.20 0.50

A8 0.8 0.60 0.25 0.50

A9 1 1 1 0.50

A10 1 1 0.25 1

2.6 Developing Weighted Sum Method (WSM) Matrix

To build the WSM matrix, each student element of the normalization matrix is calculated through the equation 3 below.

𝑆_𝑖= ∑ 𝐴_𝑖𝑗 𝐶_𝑖 (3)

Where Aij is the weight of each student with respect to the criteria used, Ci is the weight of relative importance from each criteria which can be seen in table 2 and Si is total value of the WSM method. In this case, the matrix of each element can be described as follows :

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[

𝐴₁₁ 𝐶₁ 𝐴₁₂ 𝐶₂ 𝐴₁₃ 𝐶₃ 𝐴_1𝑗 𝐶_𝑗 𝐴₂₁ 𝐶₁ 𝐴₂₂ 𝐶₂ 𝐴₁₃ 𝐶₃ 𝐴_1𝑗 𝐶_𝑗 𝐴₃₁ 𝐶₁ 𝐴₂₂ 𝐶₂ 𝐴₁₃ 𝐶₃ 𝐴_1𝑗 𝐶_𝑗

⋮ ⋮ ⋮ ⋮ 𝐴_𝑖1 𝐶₁ 𝐴_𝑖2 𝐶₂ 𝐴₁₃ 𝐶₃ 𝐴_𝑖𝑗 𝐶_𝑗]

By applying the matrix above, the value can be obtained by performing the following calculations:

A1 : A11 = (1*0.40), A12 = (0.75*0.15), A13 = (0.33*0.30), A14 = (1*0.30)

A11 = 0.40, A12 = 0.11, A13 = 0.10, A14 = 0.15

A2 : A21 = (1*0.40), A22 = (0.60*0.15), A23 = (0.25*0.30), A24 = (1*0.30)

A21 = 0.40, A22 = 0.09, A23 = 0.08, A24 = 0.15

A3 : A31 = (1*0.40), A32 = ( 0.60*0.15), A33 = (0.50*0.30), A34 = (1*0.30)

A31 = 0.40, A32 = 0.09, A33 = 0.15, A34 = 0.15

⫶ ⫶ ⫶ ⫶ ⫶

A10 : A10,1 =(1*0.40), A10,2 =(1*0.15), A10,3 =( 0.25*0.30), A10,4=(1*0.30) A10,1 = 0.40, A10,2 = 0.15, A10,3 = 0.08, A10,4 = 0.15 The results of the calculations performed on each element are shown in the table 7 below.

Table 7. Table of WSM matrix Student

Code

Criteria Code

C1 C2 C3 C4

A1 0.40 0.11 0.10 0.15

A2 0.40 0.09 0.08 0.15

A3 0.40 0.09 0.15 0.15

A4 0.40 0.09 0.06 0.15

A5 0.16 0.09 0.10 0.15

A6 0.40 0.15 0.10 0.15

A7 0.32 0.09 0.06 0.08

A8 0.32 0.09 0.08 0.08

A9 0.40 0.15 0.30 0.08

A10 0.40 0.15 0.08 0.15

After the value of all the elements of the matrix is obtained then each criterion is added up to obtain the total value of the WSM method according to equation 3.

A1 = A11 + A12 + A13 + A14

= 0.40 + 0.11 + 0.10 + 0.15

= 0.76

A2 = A21 + A22 + A23 + A24

= 0.40 + 0.09 + 0.08 + 0.15

= 0.72 A3 = A31 + A32 +A33 + A34

= 0.40 + 0.09 + 0.15 + 0.15

= 0.79

A4 = A41 + A42 +A43 + A44

= 0.16 + 0.09 + 0.10 + 0.15

= 0.70

⫶ ⫶ ⫶ ⫶

A9 = A10,1 + A10,2 + A10,3 + A10,4

= 0.40 + 0.15 + 0.30 + 0.08

= 0.93

A10 = A10,1 + A10,2 + A10,3 + A10,4

= 0.40 + 0.15 + 0.08 + 0.15

= 0.78

In Table 8 displays the results of the total value results in the above calculations.

Table 8. Table of total WSM Student

Code

Total Sum (S)

A1 0.76

A2 0.72

A3 0.79

A4 0.70

A5 0.50

A6 0.80

A7 0.55

A8 0.56

A9 0.93

A10 0.78

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DOI: 10.30865/mib.v7i1.5280

Muhadi M. Ilyas Gultom, Copyright © 2023, MIB, Page 569 The table above is the final result of the WSM method. This result is the first part to become the material of the WASPAS method where the total sum (S) column represents the total weight of each candidate in the WSM method.

2.7 Developing Weighted Product Method (WPM) Matrix

The formation of this matrix has similarities as the WSM matrix but differs in its calculations. To build the WPM matrix, the following equation is used.

𝑃_𝑖= ∏(𝐴_𝑖𝑗)^𝐶^𝑖 (4)

Where Aij is the weight of each student with respect to the criteria used, Ci is the weight of relative importance from each criteria and Pi is total value of WPM for each candidate . The matrix of each element can be described as follows :

𝐴̅ =

[

( 𝐴11)^𝐶¹ ( 𝐴₁₂)^𝐶² ( 𝐴₁₃)^𝐶³ ( 𝐴_1𝑗)^𝐶^𝑗 ( 𝐴₂₁)^𝐶¹ ( 𝐴₂₂)^𝐶² ( 𝐴₂₃)^𝐶³ ( 𝐴_1𝑗)^𝐶^𝑗 ( 𝐴₃₁)^𝐶¹ ( 𝐴₃₂)^𝐶² ( 𝐴₃₃)^𝐶³ ( 𝐴_1𝑗)^𝐶^𝑗

⋮ ⋮ ⋮ ⋮ ( 𝐴_𝑖1)^𝐶¹ ( 𝐴_𝑖2)^𝐶² ( 𝐴_𝑖3)^𝐶³ ( 𝐴_𝑖𝑗)^𝐶^𝑗 ]

From the matrix above, the calculation for each element of the matrix is carried out as follows:

A1 : A11 = (1^0.40), A12 = (0.75^0.15), A13 = (0.33^0.30), A14 = (1^0.30)

A11 = 1, A12 = 0.96, A13 = 0.72, A14 = 1

A2 : A21 = (1^0.40), A22 = (0.60^0.15), A23 = (0.25^0.30), A24 = (1^0.30)

A21 = 1, A22 = 0.93, A23 = 0.66, A24 = 1

A3 : A31 = (1^0.40), A32 = ( 0.60^0.15), A33 = (0.50^0.30), A34 = (1^0.30)

A31 = 1, A32 = 0.93, A33 = 0.81, A34 = 1

⫶ ⫶ ⫶ ⫶ ⫶

A10 : A10,1 =(1^0.40), A10,2 =(1^0.15), A10,3 =( 0.25^0.30), A10,4=(1^0.30)

A10,1 =1, A10,2 = 1, A10,3 = 0.66, A10,4 = 1

The results of the calculations performed on each element are shown in table 9 below.

Table 9. Table of WPM matrix Student

Code

Criteria Code

C1 C2 C3 C4

A1 1.00 0.96 0.72 1.00

A2 1.00 0.93 0.66 1.00

A3 1.00 0.93 0.81 1.00

A4 1.00 0.93 0.62 1.00

A5 0.69 0.93 0.72 1.00

A6 1.00 1.00 0.72 1.00

A7 0.91 0.93 0.62 0.90

A8 0.91 0.93 0.66 0.90

A9 1.00 1.00 1.00 0.90

A10 1.00 1.00 0.66 1.00

After all the elements in the matrix are weighted, each criteria is multiplied to obtain the total product value according to equation 4.

A1 = A11 * A12 * A13 * A14

= 0.40 * 0.11 * 0.10 * 0.15

= 0.69

A2 = A21 * A22 * A23 * A24

= 0.40 * 0.09 * 0.08 * 0.15

= 0.61 A3 = A31 * A32 *A33 * A34

= 0.40 * 0.09 * 0.15 * 0.15

= 0.75

A4 = A41 * A42 *A43 * A44

= 0.16 * 0.09 * 0.10 * 0.15

= 0.57

⫶ ⫶ ⫶ ⫶

A9 = A10,1 * A10,2 * A10,3 * A10,4

= 0.40 * 0.15 * 0.30 * 0.08

= 0.90

A10 = A10,1 * A10,2 * A10,3 * A10,4

= 0.40 * 0.15 * 0.08 * 0.15

= 0.6

The results of the calculations performed on each element are shown in table 10.

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Student Code Total Product (P)

A1 0.69

A2 0.61

A3 0.75

A4 0.57

A5 0.46

A6 0.72

A7 0.47

A8 0.50

A9 0.90

A10 0.66

The final result of the weighted product method is the total weight where the P column represents the total weight of each candidate for this method.

2.8 Total Joint(Q)

To generalize and integrate addition and multiplication methods, a Joint Generalized Criteria is needed [20] which is obtained by the following equation.

𝑄_𝑖= ¹

2(𝑆_𝑖+ 𝑃_𝑖) = ¹

2(∑ 𝐴̅_𝑖𝑗 𝐶_𝑖+ ∏(𝐴̅_𝑖𝑗)^𝐶^𝑖) (5)

The calculation of the above equation is carried out as follows.

Q1 = (0.76 + 0.69) * 0.5 = 0.73, Q2 = (0.72 + 0.61) * 0.5 = 0.66 Q3 = (0.79 + 0.75) * 0.5 = 0.77, Q4 = (0.70 + 0.57) * 0.5 = 0.64 Q5 = (0.50 + 0.46) * 0.5 = 0.48, Q6 = (0.80 + 0.72) * 0.5 = 0.76 Q7 = (0.55 + 0.47) * 0.5 = 0.51, Q8 = (0.56 + 0.50) * 0.5 = 0.53 Q9 = (0.93 + 0.90) * 0.5 = 0.91, Q10 = (0.78 + 0.66) * 0.5 = 0.72

The results of the above calculations are shown in table 11.

Table 11. Table of total joint Student Code Total Joint(Q) Ranking

A1 0.73 4

A2 0.66 6

A3 0.77 2

A4 0.64 7

A5 0.48 10

A6 0.76 3

A7 0.51 9

A8 0.53 8

A9 0.91 1

A10 0.72 5

This result is the value of the WASPAS as a whole method where to sort the rankings is to rank the highest value as the top rank. From the table, it can be seen that students with code A9 get the top rank while students with code A5 are in the lowest rank.

2.9 Lambda Variable (𝝀)

In this study, the lambda variable is used to increase the level of ranking accuracy for each candidate. In general the lambda variable elements are in the range of values from 1 to 0. If the lambda is set constantly using each element there will be a random change in the candidate rank. To prevent that problem, the equation below is used to find the lambda values that vary in each candidate.

𝜆_𝑖=

(∑_𝐴_̅^𝑃𝑖 𝑖𝑗)

2 (𝐴̅_𝑖𝑗)²

(∑_𝐴_̅^𝑆𝑖 𝑖𝑗)

2

(𝐴̅_𝑖𝑗)²− (∑_𝐴_̅^𝑃𝑖 𝑖𝑗)

2 (𝐴̅_𝑖𝑗)²

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The lambda value will vary with each candidate and must be calculated first before applying the total joint calculation with the following equation.

𝑄_𝑖= 𝜆(𝑆_𝑖) + (1 − 𝜆)(𝑃_𝑖) (7)

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DOI: 10.30865/mib.v7i1.5280

𝝀1 = (1.89) / (2.32 - 1.89) = 4.43, 𝝀2 = (1.49) / (2.04 + 1.49) = 2.71 𝝀3 = (2.26 / (2.49 - 2.26) = 9.74, 𝝀4 = (1.30) / (1.96 + 1.30) = 1.99 𝝀5 = (0.85) / (1 - 0.85) = 5.79, 𝝀6 = (2.06) / (2.56 + 2.06) = 4.21 𝝀7 = (0.88) / (1.18 - 0.88) = 2.95, 𝝀8 = (1.01) / (1.25 + 1.01) = 4.23 𝝀9 = (3.24) / (3.42 - 3.24) = 18.7, 𝝀10 = (1.74) / (2.40 + 1.74) = 2.63

After the lambda value is obtained, the total joint calculation is carried out using equation 7 as follows.

Q1 = (𝝀1 * S1) + ((1-𝝀1) * (P1))

= (4.43 * 0.76) + ((1 - 4.43) * 0.69)

= (3.38) + (-3.43 * 0.69)

= 1.015

Q2 = (𝝀2 * S2) + ((1-𝝀2) * (P2))

= (2.71 * 0.72) + ((1 - 2.71) * 0.61)

= (1.93) + (-1.71 * 0.61)

= 0.892

Q3 = (𝝀3 * S3) + ((1-𝝀3) * (P3))

= (9.74 * 0.79) + ((1 - 9.74) * 0.75)

= (7.69) + (-8.74 * 0.75)

= 1.119

Q4 = (𝝀4 * S4) + ((1-𝝀4) * (P4))

= (1.99* 0.70) + ((1 - 1.99) * 0.57)

= (1.39) + (-0.99 * 0.57)

= 0.828

⫶ ⫶

Q9 = (𝝀9 * S9) + ((1-𝝀9) * (P9))

= (18.7 * 0.93) + ((1 - 18.7) * 0.90)

= (17.3) + (-17.7 * 0.90)

= 1.346

⫶ ⫶

Q10 = (𝝀10 * S10) + ((1-𝝀10) * (P10)) = (2.63 * 0.78) + ((1 - 2.63) * 0.66) = (2.04) + (-1.63 * 0.66)

= 0.963

The table 12 below is the final result of the total joint on the use of the lambda variable with the same rank as the previous rule where lambda(𝝀) column represents the variation of the lambda value in each candidate .

Table 12. Table of total joint

Student Code Lambda (𝝀) Total Joint(Q) Ranking

A1 4.439 1.015 4

A2 2.710 0.892 6

A3 9.744 1.119 2

A4 1.999 0.828 7

A5 5.797 0.683 10

A6 4.215 1.059 3

A7 2.956 0.689 9

A8 4.238 0.742 8

A9 18.72 1.346 1

A10 2.632 0.963 5

From the table above, it is known that increasing accuracy does not cause changes in candidate rankings.

With this, it can be stated that the increase in accuracy works well because the ranking consistency is not at odds with the initial method before the WASPAS method was combined with the lambda variable

3. RESULT AND DISCUSSION

From the calculation results on the application of the general WASPAS method and the improved WASPAS method using lambda variables on 10 alternatives that have different attributes, the results are presented in table 13 below.

Table 13. Table of method comparison

Student Code Total Joint (Without 𝝀) Total Joint (With 𝝀) Ranking

A1 0.73 1.02 4

A2 0.66 0.89 6

A3 0.77 1.12 2

A4 0.64 0.83 7

A5 0.48 0.68 10

A6 0.76 1.06 3

A7 0.51 0.69 9

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A8 0.53 0.74 8

A9 0.91 1.35 1

A10 0.72 0.96 5

In the table above it can be seen that A9 is the alternative with the highest rank with an increase in accuracy from 0,73 to 1.02 with a percentage of accuracy of 44% ending with A5 as an alternative with the lowest increase in accuracy from 0,48 to 0,69 with a percentage of accuracy of 19%. from table 11 and table 12 it can be seen that the use of the lambda variable in the WASPAS method does not shift the ranking order to the alternatives.

Therefore the method used to improve the WASPAS method in this study does not damage the precision between alternatives. If there is an abnormality in the ranking order of the alternatives, the accuracy level cannot be considered as working properly because it is contrary to the original method. The increase in the accuracy of all alternatives can be seen in table 14 below.

Table 14. Table of method comparison Student Code Accuracy Improvement

A1 29%

A2 23%

A3 35%

A4 19%

A5 20%

A6 30%

A7 18%

A8 21%

A9 44%

A10 24%

From the table above it is known that each alternative has a different increase in accuracy but this increase does not exceed the limit of the ranking of the alternative. From the experiments in this study it is known that this method constantly provides an increase in accuracy for all alternatives above 17%. In its implementation, cases in this study are generally used to process data from a very large number of alternatives depending on the number of departments considered. Therefore we need a method with low computational complexity to provide high performance in executing all data. From equation 5 it can be seen that the WASPAS method has that advantage where the formula used only uses simple arithmetic operations and also does not use iterations which will burden the memory to carry out the calculation process.

The application of the lambda variable also has the same thing because to obtain its value it only takes the value of the two WSM (weighted sum method) and WPM (weighted product method) methods previously obtained and calculates both with a simple process as well. The possible causes for the decrease in performance are in the large weight values of the alternatives, but with the normalization and assessment stages in the sub-criteria range special provisions have been given which are usually determined by an expert which in some cases has an unfavorable impact. To be able to more easily see the comparison of the method results are shown in figure 2.

Figure 2. Accuracy comparison diagram

From the diagram above, it can be seen that the use of the lambda variable can significantly increase the accuracy of the waspas method in the case of selecting the best student.can be seen there are 3 bars that show the difference in the results of each method for each alternative. The first bar shows the results of the common WASPAS method,

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DOI: 10.30865/mib.v7i1.5280

Muhadi M. Ilyas Gultom, Copyright © 2023, MIB, Page 573 the second bar shows the WASPAS method with increased accuracy with the lambda variable and the third bar is the percentage of the increase in accuracy. From the data import of accuracy above, the average increase obtained for each candidate is 26.34%. From that result it can be concluded that there is a significant increase in accuracy for each alternative that has been tested. For more details, figure 3 displays the order of obtaining the accuracy ranking of all alternatives.

Figure 3. Rangking of accuracy diagram

One of the advantages of this method is that an expert or human can see the errors made in the calculations.

If we look at table 9, we can see that alternative A9 has the most criteria with the highest score which will affect the results of the ranking and also have the results of calculating the lambda variant with the highest value of 18.7 where it will greatly affect the increase in accuracy in calculation for equation 7. To validate errors in the application of additional methods to increase accuracy is also very easy because the ranking of alternatives must be aligned with the level of increasing accuracy.

From this study it was found that lambda which usually has a variety of values with a range of positive elements is simplified by using the optimal value of that element which is the key to increasing maximum accuracy in this study. Therefore, it can be stated that the use of the lambda variable in this study obtains maximum result in increasing the accuracy of what can be produced from combining the WASPAS method with the Lambda Variable.

4. CONCLUSION

After conducting research and analyzing the performance of the WASPAS method with and without the lambda variable, it can be concluded that the lambda variable is able to increase ranking accuracy in multi-criteria decision making in this case the selection of the best students, and also after observing the process of calculating the determination of the lambda variable, it is known that the use of a lambda variable that varies according to each alternative in this case is called a candidate has better performance than a lambda variable with constant value in each alternative. In fact, the researcher found that the constant assignment of the lambda variable can cause disturbances in the ranking of the tested alternatives. With the average value obtained, it can be concluded that the WASPAS method with the lambda variable is able to improve the performance of the WASPAS method. In the researchers observations, the equation used to obtain variations in the lambda value can be increased by changing the determinant operator as was done in this research, where the subtraction process is a modification that is proven to have an impact that is contrary to the results commonly used, although so far there has been no the finding of problems that could be detrimental due to the changes. In this study also obtained the best way to use the lambda variable where the use of optimal range values at the same time eliminates variation so that the calculation performance also increases.

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