Introduction to Fuzzy Modelling AHP Example

Due to the increasing demand for online calls and meetings, the manager of sales department in an international company producing fast-moving consumer goods needs to buy a mobile phone. He is in between buying one of the two phones: Phone-1 and Phone-2. In making his decision, he considers four attributes: the price, the storage space, camera, and looks. His comparison between the criteria (attributes) are displayed in the following table:

Table 1. Comparison matrix for criteria

CRITERIA Price Storage Space Camera Looks
Price (1,1,1) (4,5,6) (3,4,5) (6,7,8)
Storage Space (1/6,1/5,1/4) (1,1,1) (1/4,1/3,1/2) (2,3,4)
Camera (1/5,1/4,1/3) (2,3,4) (1,1,1) (2,3,4)
Looks (1/8,1/7,1/6) (1/4,1/3,1/2) (1/4,1/3,1/2) (1,1,1)

In addition, his comparisons of alternatives with respect to each alternative phone are provided in the following tables:

Table 2. Comparison of alternative phones with respect to “Price” criteria

Phone 1 Phone 2
Phone 1 (1,1,1) (2,3,4)
Phone 2 (1/4,1/3,1/2) (1,1,1)

 

Table 3. Comparison of alternative phones with respect to “Storage Space” criteria

Phone 1 Phone 2
Phone 1 (1,1,1) (6,7,8)
Phone 2 (1/8,1/7,1/6) (1,1,1)

 

Table 4. Comparison of alternative phones with respect to “Camera” criteria

Phone 1 Phone 2
Phone 1 (1,1,1) (1/6,1/5,1/4)
Phone 2 (4,5,6) (1,1,1)

 

Table 5. Comparison of alternative phones with respect to “Looks” criteria

Phone 1 Phone 2
Phone 1 (1,1,1) (1/7,1/6,1/5)
Phone 2 (5,6,7) (1,1,1)

 

In this example, best mobile phone that suits the manager’s criteria is found considering fuzzy AHP method.

Firstly, weights of criteria will be determined. The geometric mean of fuzzy comparison values of each criterion is calculated by the equation below.

Thus, the geometric means of fuzzy comparison values of all criteria are shown in Table 6.

Table 6. Geometric means for all criteria

CRITERIA 𝑟̃𝑖
Price 2.91 3.44 3.94
Storage Space 0.54 0.67 0.84
Camera 0.95 1.22 1.52
Looks 0.30 0.35 0.45
Total 4.7 5.68 6.75
Reverse 0.21 0.18 0.15
Increasing Order 0.15 0.18 0.21

 

In the next step, the fuzzy weight of each criterion 𝑤̃𝑖 is with equation below.

𝑤̃𝑖 = 𝑟̃𝑖 ⊗ (𝑟̃1 ⊕ 𝑟2 ⊕ … 𝑟̃𝑛)−1 = (𝑙𝑤𝑖, 𝑚𝑤𝑖, 𝑢𝑤𝑖)

𝑤̃1  = [ (2.91 ∗ 0.15); (3.44 ∗ 0.18); (3.94 ∗ 0.21) = [0.437; 0.619; 0.827]

𝑤̃2  = [ (0.54 ∗ 0.15); (0.67 ∗ 0.18); (0.84 ∗ 0.21) = [0.081; 0.121; 0.176]

𝑤̃3  = [ (0.95 ∗ 0.15); (1.22 ∗ 0.18); (1.52 ∗ 0.21) = [0.143; 0.220; 0.319]

𝑤̃4 = [ (0.30 ∗ 0.15); (0.35 ∗ 0.18); (0.45 ∗ 0.21) = [0.045; 0.063; 0.095]

 

Table 7. Relative fuzzy weights of each criterion

CRITERIA 𝑤̃𝑖
Price 0.437 0.619 0.827
Storage Space 0.081 0.121 0.176
Camera 0.143 0.220 0.319
Looks 0.045 0.063 0.095

 

In the next step, the relative non- fuzzy weight of each criterion (𝑀𝑖) is calculated by taking the average fuzzy numbers for each criterion.

𝑀𝑖 = 𝑙𝑤𝑖, 𝑚𝑤𝑖, 𝑢𝑤𝑖 3

Then, by using non fuzzy weights, the normalized weights of each criterion (𝑁𝑖) are calculated and showed in Table 8.

Table 8. Averaged and normalized relative weights

CRITERIA 𝐌𝐢 𝐍𝐢
Price 0.628 0.599
Storage Space 0.126 0.120
Camera 0.227 0.217
Looks 0.068 0.065
Totals 1.05 1.00

 

For the “Price, Storage Space, Camera and Looks” attributes, same steps are applied. 𝑟̃𝑖, 𝑤̃𝑖,

Mi, Ni values are calculated respectively for each 4 criteria.

Price Criteria

Table 2. Comparison of alternative phones with respect to Price criteria

Phone 1 Phone 2
Phone 1 (1,1,1) (2,3,4)
Phone 2 (1/4,1/3,1/2) (1,1,1)

 

Table 9. Geometric means of alternatives with respect to Price Criterion

ALTERNATIVES 𝑟̃𝑖
Phone 1 1.41 1.73 2.00
Phone 2 0.50 0.58 0.71
Total 1.91 2.31 2.71
Reverse 0.524 0.433 0.369
Increasing Order 0.369 0.433 0.524

 

𝑤̃1  = [ (1.41 ∗ 0.369); (1.73 ∗ 0.433); (2.00 ∗ 0.524) = [0.520; 0.749; 1.048]

𝑤̃2 = [ (0.50 ∗ 0.369); (0.58 ∗ 0.433); (0.71 ∗ 0.524) = [0.185; 0.251; 0.372]

 

Table 10. Fuzzy weights of alternatives with respect to Price Criteria

ALTERNATIVES 𝑤̃𝑖
Phone 1 0.520 0.749 1.048
Phone 2 0.185 0.251 0.372

 

Table 11. Averaged and normalized relative weights for Price Criteria

ALTERNATIVES 𝐌𝐢 𝐍𝐢
Phone 1 0.772 0.742
Phone 2 0.269 0.258
Totals 1.041 1

 

Storage Space Criteria

Table 3. Comparison of alternative phones with respect to Storage Space criteria

Phone 1 Phone 2
Phone 1 (1,1,1) (6,7,8)
Phone 2 (1/8,1/7,1/6) (1,1,1)

Table 12. Geometric means of alternatives with respect to Storage Space Criterion

ALTERNATIVES 𝑟̃𝑖
Phone 1 2.45 2.65 2.83
Phone 2 0.35 0.38 0.41
Total 2.80 3.03 3.24
Reverse 0.36 0.33 0.31
Increasing Order 0.31 0.33 0.36

 

𝑤̃1  = [ (2.45 ∗ 0.31); (2.65 ∗ 0.33); (2.83 ∗ 0.36) = [0.760; 0.875; 1.019]

𝑤̃2= [ (0.35 ∗ 0.31); (0.38 ∗ 0.33); (0.41 ∗ 0.36) = [0.109; 0.125; 0.178]

 

Table 13. Fuzzy weights of alternatives with respect to Storage Space Criteria

ALTERNATIVES 𝑤̃𝑖
Phone 1 0.760 0.875 1.019
Phone 2 0.109 0.125 0.178

 

Table 14. Averaged and normalized relative weights for Storage Space Criteria

ALTERNATIVES 𝐌𝐢 𝐍𝐢
Phone 1 0.885 0.866
Phone 2 0.137 0.134
Totals 1.022 1

 

Camera Criteria

Table 4. Comparison of alternative phones with respect to “Camera” criteria

Phone 1 Phone 2
Phone 1 (1,1,1) (1/6,1/5,1/4)
Phone 2 (4,5,6) (1,1,1)

Table 15. Geometric means of alternatives with respect to Camera Criterion

ALTERNATIVES 𝑟̃𝑖
Phone 1 0.41 0.45 0.5
Phone 2 2 2.24 2.45
Total 2.41 2.69 2.95
Reverse 0.41 0.37 0.34
Increasing Order 0.34 0.37 0.41

 

𝑤̃1  = [ (0.41 ∗ 0.34); (0.45 ∗ 0.37); (0.5 ∗ 0.41) = [0.140; 0.167; 0.205]

𝑤̃2 = [ (2 ∗ 0.34); (2.24 ∗ 0.37); (2.45 ∗ 0.41) = [0.68; 0.829; 1.005]

 

Table 16. Fuzzy weights of alternatives with respect to Camera Criteria

ALTERNATIVES 𝑤̃𝑖
Phone 1 0.140 0.167 0.205
Phone 2 0.68 0.829 1.005

 

Table 17. Averaged and normalized relative weights for Camera Criteria

ALTERNATIVES 𝐌𝐢 𝐍𝐢
Phone 1 0.171 0.169
Phone 2 0.838 0.831
Totals 1.009 1

Look Criteria

Table 5. Comparison of alternative phones with respect to “Looks” criteria

Phone 1 Phone 2
Phone 1 (1,1,1) (1/7,1/6,1/5)
Phone 2 (5,6,7) (1,1,1)

Table 18. Geometric means of alternatives with respect to Look Criterion

ALTERNATIVES 𝑟̃𝑖
Phone 1 0.38 0.41 0.45
Phone 2 2.24 2.45 2.65
Total 2.62 2.86 3.1
Reverse 0.38 0.35 0.32
Increasing Order 0.32 0.35 0.38

 

𝑤̃1  = [ (0.38 ∗ 0.32); (0.41 ∗ 0.35); (0.45 ∗ 0.38) = [0.122; 0.144; 0.171]

𝑤̃2 = [ (2.24 ∗ 0.32); (2.45 ∗ 0.35); (2.65 ∗ 0.38) = [0.717; 0.858; 1.007]

 

Table 19. Fuzzy weights of alternatives with respect to Look Criteria

ALTERNATIVES 𝑤̃𝑖
Phone 1 0.122 0.144 0.171
Phone 2 0.717 0.858 1.007

 

Table 20. Averaged and normalized relative weights for Look Criteria

ALTERNATIVES 𝐌𝐢 𝐍𝐢
Phone 1 0.146 0.145
Phone 2 0.861 0.855
Totals 1.007 1

 

After finding the normalized non-fuzzy relative weights of each alternative for those 4 criteria, the found data entered in table like below.

 

Table 21. Normalized non-fuzzy relative weights of each alternative for each criterion

ALTERNATIVES PRICE STORAGE SPACE CAMERA LOOKS
PHONE 1 0.742 0.866 0.169 0.145
PHONE 2 0.258 0.134 0.831 0.855

 

By using Table 7 and Table 21, necessary calculations are made and individual scores of each alternative for each criterion are presented in Table 22.

Table 22. Aggregated results for each alternative according to each criterion

CRITERIA Scores of Alternatives with respect to related Criterion
Weights Phone 1 Phone 2
Price 0.628 0.742 0.258
Storage Space 0.126 0.866 0.134
Camera 0.227 0.169 0.831
Looks 0.068 0.145 0.855
Totals 0.623 0.426

 

𝐹𝑜𝑟 𝑃ℎ𝑜𝑛𝑒 1 = 0.628 ∗ 0.742 + 0.126 ∗ 0.866 + 0.227 ∗ 0.169 + 0.068 ∗ 0.145 = 0.623

𝐹𝑜𝑟 𝑃ℎ𝑜𝑛𝑒 2 = 0.628 ∗ 0.258 + 0.126 ∗ 0.134 + 0.227 ∗ 0.831 + 0.068 ∗ 0.855 = 0.426

Depending on this result, Phone-1 as an alternative has the largest total score. Therefore, it is suggested as the best phone among two of them, with respect to 4 criteria and the fuzzy preferences of decision makers.

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