Q5Soft Computing
Question
Q.5. The following data was determined by the pairwise comparison of work preferences of 100 people: When it was compared with software (S), 40 persons polled preferred hardware (H), 38 of them preferred teaching (T), 48 of them preferred business (B) and 30 preferred textiles (TX). In comparison with hardware (H), the preferences were 60 for S, 52 for T, 53 for B and 35 for TX. When compared with teaching, the preferences were 62 for S, 48 for H, 65 for B and 56 for TX. In comparison with business, the preferences were 52 for S, 47 for H, 35 for T and 60 for TX. When compared with textile, the preferences were 70 for S, 65 for H, 44 for T and 40 for B. Using the rank ordering, plot the membership function for the 'most preferred work.'
Answer
The rank ordering method determines fuzzy membership values from pairwise comparison (preference) data collected from a group of respondents, converting subjective, pairwise preference judgments into a single, consistent membership function over the set of alternatives being compared, based on the principle that an alternative preferred more often (and more strongly) across all pairwise comparisons should receive a correspondingly higher membership degree in the fuzzy set representing the target linguistic concept, here 'most preferred work.'
Constructing the Pairwise Preference Matrix
The given data specifies, for each pair of alternatives, how many of the 100 polled people preferred one over the other. Organizing this into a complete pairwise preference matrix (where each entry represents the number of people preferring the row alternative over the column alternative, out of the total 100 respondents), using the given data: comparing S against others, 40 preferred H over S, 38 preferred T over S, 48 preferred B over S, 30 preferred TX over S, meaning S's own preference count against each is the complement out of 100 in each specific pairwise poll (60 preferred S over H, 62 preferred S over T, 52 preferred S over B, 70 preferred S over TX, using the complementary data given directly for the H, T, B, and TX rows).
| vs S | vs H | vs T | vs B | vs TX | Row Sum |
|---|---|---|---|---|---|
| S: - | 40 | 38 | 48 | 30 | 156 |
| H: 60 | - | 52 | 53 | 35 | 200 |
| T: 62 | 48 | - | 65 | 56 | 231 |
| B: 52 | 47 | 35 | - | 60 | 194 |
| TX: 70 | 65 | 44 | 40 | - | 219 |
Each row sum is computed by adding all the entries in that row (the number of times, across all four pairwise comparisons involving that alternative, that this alternative was preferred): Row sum for S = 40+38+48+30 = 156; Row sum for H = 60+52+53+35 = 200; Row sum for T = 62+48+65+56 = 231; Row sum for B = 52+47+35+60 = 194; Row sum for TX = 70+65+44+40 = 219.
Computing Membership Values
The membership degree of each alternative in the fuzzy set 'most preferred work' is computed by normalizing each alternative's row sum by the grand total of all row sums (1000 in this case, since each of the 5 alternatives contributes 4 pairwise comparisons of up to 100 each, giving a maximum possible total of 54100/2 = 1000 when counted this way, consistent with the computed total above): mu(alternative) = row_sum(alternative) / total_of_all_row_sums.
| Alternative | Row Sum | Membership mu |
|---|---|---|
| Software (S) | 156 | 0.156 |
| Hardware (H) | 200 | 0.200 |
| Teaching (T) | 231 | 0.231 |
| Business (B) | 194 | 0.194 |
| Textiles (TX) | 219 | 0.219 |
Membership Function Plot
The resulting membership function, plotted as a bar chart across the five alternatives, shows Teaching (T) with the highest membership value (0.231) in the fuzzy set 'most preferred work,' followed closely by Textiles (0.219), Hardware (0.200), Business (0.194), and finally Software (0.156) with the lowest membership degree among the five alternatives; this ranking reflects the aggregate outcome of all pairwise preference comparisons collected from the 100 respondents, converted through the rank ordering method into a properly normalized fuzzy membership function whose values sum to 1.0 across all five alternatives (0.156+0.200+0.231+0.194+0.219 = 1.000), consistent with the requirement that a properly normalized membership distribution over a complete, mutually exclusive set of alternatives sum to unity. This rank-ordering-derived membership function can then be used directly as a fuzzy set input to further fuzzy decision-making or multi-criteria analysis procedures, wherever a numerical representation of aggregate group preference (as opposed to a simple majority-vote ranking, which would discard the relative strength of preference information captured by the raw pairwise comparison counts) is required.