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.