Weighted Product (WP) method is one of the solution methods in decision support systems. This method evaluates several alternatives against a set of attributes or criteria, where each attribute is independent of each other.
Weighted Product (WP) Method
According to Yoon (in Kusumadewi's book, 2006), the weighted product method uses a multiplication technique to connect attribute ratings, where the rating of each attribute must first be raised to the power of the weight of the attribute in question.
The steps for completing WP are as follows:
1. Determine the criteria
These are the criteria that will be used as a reference in decision making, namely Ci and the nature of each criterion.
2. Determine the suitability rating
That is, rating the suitability of each alternative on each criterion, and creating a decision matrix.
3. Perform weight normalization
Normalized Weight = Weight of each criterion / sum of all criterion weights.
The value of the total weight must satisfy the equation:
4. Determine the value of the vector S
By multiplying all the criteria for an alternative with the weight as a positive exponent for the benefit criteria and the weight functions as a negative exponent for the cost criteria.
The formula for calculating the preference value for alternative Ai is given as follows:
Information:
- S : represents the alternative preferences which are analogous to the vector S.
- x : indicates the criteria value
- w : indicates the weight of the criteria
- i : states alternative
- j : stating the criteria
- n : indicates the number of criteria
5. Determine the value of the vector V
This is the value that will be used for ranking.
The relative preference value of each alternative can be calculated using the formula:
Information:
- V : represents the alternative preference which is analogous to the vector V
- x : indicates the criteria value
- w : indicates the weight of the criteria
- i : states alternative
- j : stating the criteria
- n : indicates the number of criteria
6. Ranking Vector Values V
As well as making conclusions as the final stage.
Case Study Example
(Quoting from the book by Sri Kusumadewi, et al., 2006):
A company in the Special Region of Yogyakarta (DIY) wants to build a warehouse that will be used as a place to temporarily store its production results.
There are 3 locations that will be alternatives, namely:
- A1 = Climbing,
- A2 = Kalasan,
- A3 = Big City.
There are 5 criteria that are used as a reference in decision making, namely: decision making, namely:
- C1 = distance to the nearest market (km),
- C2 = population density around the location (people/km2);
- C3 = distance from factory (km);
- C4 = distance to existing warehouse (km);
- C5 = land price for the location (1,000,000 Rp/m2).
The level of importance (weight) of each criterion is also assessed from 1 to 5, namely:
- = Very low,
- = Low,
- = Enough,
- = Height,
- = Very High.
The decision maker assigns preference weights to each criterion as follows:
Completion
Using the Weighted Product (WP) method of solving steps.
1. Determine the criteria
2. Determine the suitability rating
3. Perform weight normalization
W = (5, 3, 4, 4, 2)
So the weight corrections made are:
W1 = 5/(5+3+4+4+2) = 5/18 = 0.28
W2 = 3/(5+3+4+4+2) = 3/18 = 0.17
W3 = 4/ (5+3+4+4+2) = 4/18 = 0.22
W4 = 4/(5+3+4+4+2) = 4/18 = 0.22
W5 = 2/(5+3+4+4+2) = 2/18 = 0.11
If the values W1+W2+W3+W4+W5 are added together, the result will be ≈ 1
W1 + W2 + W3 + W4 + W5 = 0.28 + 0.17 + 0.22 + 0.22 + 0.11 = 1
4. Determine the value of the vector S
Determine the value of the vector S by multiplying all the criteria for an alternative with the weight as a positive (+) power for the benefit criteria and the weight functions as a negative (-) power for the cost criteria.
S1 = (0.75-0.28) x (2000-0.17) x (18-0.22) x (500.22) x (500-0.11) = 0.1920
S2 = (0, 5-0.28) x (1500-0.17) x (20-0.22) x (400.22) x (450-0.11) = 0.2120
S2 = (0.9-0.28) x (2050-0.17) x (35-0.22) x (350.22) x (800-0, 11) = 0.1375
5. Determine the value of the vector V
V1 = S1 / S1 + S2 + S3
V1 = 0.1920 / 0.1920 + 0.2120 + 0.1375
V1 = 0.1920 / 0.5415
V1 = 0.3546
V2 = S2 / S1 + S2 + S3
V2 = 0.2120 / 0.1920 + 0.2120 + 0.1375
V2 = 0.2120 / 0.5415
V2 = 0.3916
V3 = S3 / S1 + S2 + S3
V3 = 0.1375 / 0.1920 + 0.2120 + 0.1375
V3 = 0.1375 / 0.5415
V3 = 0.2539
6. Ranking Vector Values V
By looking at point number 5, it can be concluded that the value of v2 is greater than the values of v1 and v3.
- Rank 1 -> v2 = 0.3916
- Rank 2 -> v1 = 0.3546
- Rank 3 -> v3 = 0.2539
EXERCISE
Weighted Product (WP) Method Problem Solving
Reference: 3rd Meeting Module Weighted Product (WP), compiled by Andri Syafrianto, S.Kom., M.Cs.
In the discussion of this question, the Criteria table and the Match table have been identified, so in the discussion of this question, we have been helped with the first 2 steps of the solution, so we just have to continue until we reach the final step, namely the conclusion/selection of the ranking results.
Weighted Product (WP) Method Problem Solving
1. Determine the criteria that will be used as a reference in decision making, namely Ci and the nature of each criterion.
| KRITERIA | SIFAT | BOBOT |
|-----------------------------------------------------------------|---------|-------|
| C1: Kesesuaian proposal yang diajukan terhadap persyaratan PNPM | Benefit | 5 |
| C2: Kegiatan yang diajukan mendesak untuk dilakukan | Benefit | 4 |
| C3: Pendapatan per tahun masyarakat | Cost | 4 |
| C4: Lokasi desa dilihat dari jarak dengan pusat pemerintahan | Benefit | 3 |
| C5: Tingkat kemajuan desa | Benefit | 5 |2. Determine the suitability rating of each alternative for each criterion.
| ALTERNATIF | KRITERIA | | | | |
|------------|----------|----|-------------|--------|----|
| | C1 | C2 | C3(Rp) | C4(km) | C5 |
| Sumber | 5 | 5 | 1.000.000,- | 20 | 5 |
| Sariharjo | 5 | 5 | 800.000,- | 22 | 5 |
| Sinduharjo | 5 | 3 | 850.000,- | 25 | 5 |
| Windusari | 3 | 5 | 900.000,- | 23 | 5 |
| Mranggen | 5 | 3 | 1.050.000,- | 24 | 5 |Make it a matrix;
| | | | | |
|---|---|-------------|----|---|
| 5 | 5 | 1.000.000,- | 20 | 5 |
| 5 | 5 | 800.000,- | 22 | 5 |
| 5 | 3 | 850.000,- | 25 | 5 |
| 3 | 5 | 900.000,- | 23 | 5 |
| 5 | 3 | 1.050.000,- | 24 | 5 |3. Perform weight normalization
w = (5, 4, 4, 3, 5)
w1 = 5 / (5 + 4 + 4 + 3 + 5) = 5 / 21 = 0.238095238
w2 = 4 / (5 + 4 + 4 + 3 + 5) = 4 / 21 = 0.19047619
w3 = 4 / (5 + 4 + 4 + 3 + 5) = 4 / 21 = 0.19047619
w4 = 3 / (5 + 4 + 4 + 3 + 5) = 3 / 21 = 0.142857143
w5 = 5 / (5 + 4 + 4 + 3 + 5) = 5 / 21 = 0.238095238
If the values w1+w2+w3+w4+w5 are added together, the result will be ≈ 1
4. Determine the value of the vector S
By multiplying all the criteria for an alternative with the weight as a positive exponent for the benefit criteria and the weight as a negative exponent for the cost criteria.
If the derivative formula is obtained:
S1 = C1w1 * C2w2 * C3-w3 * C4w4 * C5w5
So the final result is;
| | | | | | | | |
|-----|-------------|-------------|-------------|-------------|-------------|---|-------------|
| s1= | 1.466970666 | 1.358742449 | 0.071968567 | 1.534127405 | 1.466970666 | = | 0.322837732 |
| s2= | 1.466970666 | 1.358742449 | 0.075093424 | 1.555158537 | 1.466970666 | = | 0.341473164 |
| s3= | 1.466970666 | 1.232764839 | 0.074231267 | 1.583819609 | 1.466970666 | = | 0.311900223 |
| s4= | 1.298973522 | 1.358742449 | 0.073427471 | 1.565065608 | 1.466970666 | = | 0.297543173 |
| s5= | 1.466970666 | 1.232764839 | 0.071302835 | 1.574610106 | 1.466970666 | = | 0.297853654 |5. Determine the value of the vector V
The value of the vector V that will be used for ranking can be calculated in the following way:
If the derivative formula is obtained Vn = Sn / Total
| Maka: | S1 | S TOTAL | Hasil |
|-------|-------------|-------------|-------------|
| V1 = | 0.322837732 | 1.571607946 | 0.205418745 |
| V2 = | 0.341473164 | 1.571607946 | 0.217276303 |
| V3 = | 0.311900223 | 1.571607946 | 0.198459306 |
| V4 = | 0.297543173 | 1.571607946 | 0.189324044 |
| V5 = | 0.297853654 | 1.571607946 | 0.189521601 |6. Ranking Vector Values V
By looking at point number 5, it can be concluded that the value of V2 is greater than the value of V1.
- Rank 1 = V2
- Rank 2 = V1
Attachment: Excel Formula for Weighted Product (WP) Method
Reference:
3rd Meeting Module Weighted Product (WP), compiled by Andri Syafrianto, S.Kom., M.Cs