Extension card · m-Polar

m-Polar Linguistic TOPSIS (Adeel, Akram and Koam, 2019)

This is the form of TOPSIS for situations where a linguistic term is assessed from several independent viewpoints (poles) and built as a group decision from several decision-makers' shared opinion.

Base method

TOPSIS →

Philosophy, mechanics, strengths and weaknesses are on the base method card; this card describes only the difference.

Data type (family)

m-Polar →

What this data type is, when to use it, how to write it in a cell: the family's full account is here.

What Changes from the Base Method?

Four things change; the decision logic stays the same.

Cells. In crisp TOPSIS every cell is a single number. Here every cell is the set of degrees a linguistic term receives from m separate viewpoints (poles); every pole is a number between 0 and 1. In a model's "appearance" assessment, for instance, the value of the term "quite pretty" is scored separately on the poles of facial features, posture and body proportions. Where several decision-makers are involved, each supplies their own matrix; DecisionMind merges these into a single matrix by simple arithmetic mean, and this merging is done BEFORE weighting. Criterion weights, too, are taken separately from every decision-maker and merged the same way, by averaging.

Scale equalisation. There is no column normalisation here, unlike in crisp TOPSIS. Every pole value is already entered as a degree between 0 and 1. A raw score, for instance a mark out of 100, is not scaled by the engine; the user must first bring it into the 0-to-1 range themselves.

Weighting. In crisp TOPSIS, weighting is applied to the whole column. Here the merged weight is multiplied directly, as a scalar, into every pole of the merged matrix. This multiplication is not exponential; because there is no hesitancy layer, it differs from the exponential weighting used in MHF-TOPSIS and MPF-HF-TOPSIS.

Distance and the result. The ideal and anti-ideal are built, for every linguistic term, from the best and worst value at every pole separately. Distance is an equally weighted Euclidean distance taken over the poles; because the criterion weight is already embedded into the values at the previous step, it is not multiplied a second time in the distance formula. The closeness coefficient is again a single number between 0 and 1.

DecisionMind fixes, for this extension, merging the group before weighting and taking an equally weighted distance over the poles.

How to Read the Output

The closeness coefficient is read as in crisp TOPSIS, only it ranks this particular set of alternatives. The difference is here: the decision-makers' matrices are merged by averaging, before weighting. Because of this, a single outlying decision-maker can shift the ideal and anti-ideal point for every alternative.

Thus instead of writing:

"m-polar linguistic TOPSIS is free of subjectivity because it uses a group decision"

the report should read:

"The decision-makers' matrices have been merged by simple averaging; a single outlying decision-maker can shift the ideal and anti-ideal point, and this risk should be assessed in the report"

When to Prefer This over the Base Method

This extension is suitable where several decision-makers assess a criterion with linguistic terms from m separate viewpoints. Even with a single decision-maker, the pole structure can still be used. Where there is only a single viewpoint and the decision-maker chooses a single linguistic term, 2-tuple linguistic TOPSIS is sufficient; where the term is distributed with a probability, Probabilistic linguistic TOPSIS is sufficient; the pole layer adds unnecessary complexity in these cases. Where a criterion is measured, the base method should be kept; DecisionMind asks that the table hold a single data type.

Mistakes Specific to This Extension

Giving a raw score without scaling it. Writing a mark out of 100 directly into a cell breaks the 0-to-1 range constraint; the method does not perform this scaling itself.

Not keeping the number and meaning of poles fixed. Every pole, for instance facial features or posture, must carry the same meaning for all decision-makers, alternatives and linguistic terms; otherwise building the column-wise best and worst value is inconsistent.

Giving linguistic weight labels without converting them to numbers. Where weights are given only as labels ("medium," "high"), a numerical equivalent must first be assigned, and these must sum to 1; the method does not perform this conversion itself.

Multiplying the criterion weight a second time in the distance formula. The weight is already embedded in the weighted matrix; multiplying the deviation by the weight again in the distance calculation distorts the paper's figures.

The governing principle is this:

The decision-makers' matrices are merged before weighting; once merged, a single outlying opinion's effect becomes permanent for every alternative, and this risk must be assessed separately.

Cases

The first case is the model-appearance ranking example from Adeel, Akram and Koam (2019); the figures are the paper's own and have been independently recomputed with DecisionMind's engine. The second case is an illustrative construction.

1. Appearance assessment: Ranking five models (Adeel, Akram and Koam, 2019)

Four decision-makers assess five models against four linguistic terms of the "appearance" feature: not very attractive, fairly cute, quite pretty, very beautiful. Every term's value is scored separately on three independent poles (facial features, posture, body proportions). The four decision-makers' matrices have been merged by averaging; the table below is this merged matrix. Weights (the average of the four decision-makers): not very attractive 0.1925, fairly cute 0.2300, quite pretty 0.2596, very beautiful 0.3180.

Every cell is given as three values, one per pole, in sequence and separated by a slash.

Model	Not very attractive	Fairly cute	Quite pretty	Very beautiful
Am1	0.215 / 0.3075 / 0.235	0.4325 / 0.5375 / 0.6225	0.72 / 0.6925 / 0.7325	0.8975 / 0.92 / 0.9475
Am2	0.29 / 0.235 / 0.225	0.415 / 0.49 / 0.6975	0.6825 / 0.7375 / 0.7525	0.8775 / 0.895 / 0.9025
Am3	0.285 / 0.2625 / 0.1625	0.4875 / 0.4625 / 0.64	0.6325 / 0.625 / 0.705	0.835 / 0.815 / 0.8925
Am4	0.215 / 0.2125 / 0.235	0.345 / 0.58 / 0.71	0.6 / 0.685 / 0.7575	0.9075 / 0.87 / 0.9175
Am5	0.2 / 0.2575 / 0.2925	0.355 / 0.5 / 0.71	0.5425 / 0.595 / 0.8	0.9025 / 0.89 / 0.9125
Direction	higher is better	higher is better	higher is better	higher is better

The method multiplies every pole by its weight, builds the ideal and anti-ideal model pole by pole for every linguistic term, and computes every model's closeness coefficient.

Model	Closeness coefficient	Rank
Am1	0.668	1
Am2	0.627	2
Am4	0.494	3
Am5	0.422	4
Am3	0.377	5

The result reads as follows. Am1 holds the highest values on all three poles of "very beautiful," the heaviest term. Am2 follows immediately behind; the gap between them is 0.041. Am3 scored relatively high on "not very attractive" and low on the other terms, and finishes last.

The decision-makers' hesitation is this. If the weight of "not very attractive" is raised from 0.1925 to 0.45 and the weight of "very beautiful" is lowered from 0.318 to 0.06, the order of Am4 and Am5 reverses: Am5 comes third with 0.459, Am4 fourth with 0.423. Am1 and Am2 stay first and second. This shows that the order of the third- and fourth-place models is sensitive to the weight distribution.

In the report: "With the weights given, Am1 has the highest closeness coefficient (0.668) and, although the gap to Am2 (0.627) is small, the ranking is robust. The order between Am4 and Am5 is sensitive to the weight distribution; Am5 moves ahead if the weight of 'not very attractive' is raised noticeably."

Source: Adeel, Akram and Koam (2019), §3.1, Tables 4–8, pp. 8–11. The closeness coefficients are taken from the paper's own table, and have been verified by independently re-running DecisionMind's MPF-TOPSIS-LING engine. For Am5, the value the engine produces differs slightly from the paper's published value, though the rank is unchanged; see the approval notes for detail.

2. Publishing: Choosing among three book-cover designs

A publishing house will choose one of three cover-design proposals for a new novel. The linguistic terms are "does not catch the eye," "catches the eye moderately," and "catches the eye strongly"; every term's value has been scored separately on three reader-segment poles (young adult, middle-aged, experienced reader). Three editors' assessments have been merged by averaging.

The method computes every design's distance to the ideal and anti-ideal design pole by pole, and converts it into a closeness coefficient. Suppose the design with the highest score in the young-adult pole scored lowest in the experienced-reader pole, and still finished first, because it stood out on "catches the eye strongly," the heaviest term.

The publishing house's hesitation is this: one of the editors gave a noticeably higher score than the others in the young-adult segment. If this editor's score is removed and the average recomputed, the first-place design's lead could weaken.

In the report: "The first-place design reaches high eye-catching value through its strong score in the young-adult segment. One editor's score in this segment is noticeably higher than the others; sensitivity by editor should be checked."

3. What Not to Do

In the first case, replacing one of the poles' values in Am3's "quite pretty" term with a single decision-maker's score instead of the average of the four breaks the merging rule, and the result no longer reflects the group decision. The second mistake is writing a report without stating the weight distribution; the order between Am4 and Am5 is sensitive to the weight distribution and reverses under one scenario. The third mistake is applying the pole weight separately to both the weighted matrix and the distance formula; this counts the same weight twice and distorts the paper's figures.

Formulae

The computation step by step, with the formula behind each step.

1. Ingest the aggregated m-PF linguistic decision matrix d'^i_jk (Table 3 / Table 5 of the source: p alternatives x q linguistic values x m poles, every coordinate in [0,1]) and the aggregated linguistic-value weights w'_k. The per-decision-maker mean of Sec. 3, p. 7 is taken upstream: the engine receives only the aggregated matrix and weights. Benefit direction only; native m-polar membership vectors, no scoring or defuzzification.
$d_{j k}^{' i} \in [0, 1], i = 1, \dots, m; w_{k}^{'} \in (0, 1], k = 1 \sum q w_{k}^{'} = 1$
2. Aggregated decision matrix and weights of the source (Sec. 3, p. 7: d'^i_jk = (1/r) sum_l d^{l,i}_jk and w'_k = (1/r) sum_l w^l_k). The mean over the r decision-makers is not recomputed here; the supplied matrix and weights are taken as these aggregates, the weights are normalised to sum 1, and both are published.
$d_{j k}^{' i} = \frac{1}{r} l = 1 \sum r d_{j k}^{l, i}, i = 1, \dots, m; j = 1, \dots, p; k = 1, \dots, q . w_{k}^{'} = \frac{1}{r} l = 1 \sum r w_{k}^{l}, k = 1, \dots, q; k = 1 \sum q w_{k}^{'} = 1.$
3. Form the weighted aggregated m-PF linguistic decision matrix E = [(e^1_jk, e^2_jk, …, e^m_jk)]_{p×q} by multiplying each pole component of the aggregated cell d'^i_jk by the aggregated weight w'_k of its linguistic value (Sec. 3, bottom-of-p. 7 formula and Table 7): e^i_jk = w'_k d^i_jk. The result is an m-PF number with every coordinate in [0,1] because w'_k ∈ (0,1] and d^i_jk ∈ [0,1].
$E = [(e_{j k}^{1}, e_{j k}^{2}, \dots, e_{j k}^{m})]_{p \times q}, e_{j k}^{i} = w_{k}^{'} d_{j k}^{i}, i = 1, \dots, m; j = 1, \dots, p; k = 1, \dots, q .$
4. Extract the m-PF linguistic positive ideal solution mPIS and the m-PF linguistic negative ideal solution mNIS column-wise per pole (Eqs. 1 and 2). For every linguistic value V_k (column) and every pole i, set (d^i_k)+ = max_j e^i_jk and (d^i_k)− = min_j e^i_jk. The mPIS / mNIS are q-tuples of m-PF numbers: mPIS = {((d^1_1)+,…,(d^m_1)+), …, ((d^1_q)+,…,(d^m_q)+)}; mNIS = {((d^1_1)−,…,(d^m_1)−), …, ((d^1_q)−,…,(d^m_q)−)}. All criteria are benefit-direction, so max → ideal positive and min → ideal negative everywhere.
$mPIS = {((d_{k}^{1})^{+}, (d_{k}^{2})^{+}, \dots, (d_{k}^{m})^{+})}_{k = 1}^{q}, (d_{k}^{i})^{+} = j = 1, \dots, p max e_{j k}^{i}; mNIS = {((d_{k}^{1})^{-}, (d_{k}^{2})^{-}, \dots, (d_{k}^{m})^{-})}_{k = 1}^{q}, (d_{k}^{i})^{-} = j = 1, \dots, p min e_{j k}^{i} .$
5. Compute the m-PF linguistic Euclidean distance of each alternative a_j from mPIS and from mNIS by the per-pole-equal-weighted Euclidean metric over q linguistic values and m poles (Eqs. 3 and 4): D_e(a_j, mPIS) = √((1/m) Σ_{k=1}^{q} [(e^1_jk − (d^1_k)+)^2 + (e^2_jk − (d^2_k)+)^2 + … + (e^m_jk − (d^m_k)+)^2]); D_e(a_j, mNIS) = √((1/m) Σ_{k=1}^{q} [(e^1_jk − (d^1_k)−)^2 + (e^2_jk − (d^2_k)−)^2 + … + (e^m_jk − (d^m_k)−)^2]). The (1/m) prefactor inside the square root applies per-pole-equal weighting; q linguistic values are summed without per-criterion reweighting because the criterion weights w'_k were already embedded into e^i_jk in F3.
$D_{e} (a_{j}, mPIS) = \frac{1}{m} k = 1 \sum q [(e_{j k}^{1} - (d_{k}^{1})^{+})^{2} + (e_{j k}^{2} - (d_{k}^{2})^{+})^{2} + \dots + (e_{j k}^{m} - (d_{k}^{m})^{+})^{2}]; D_{e} (a_{j}, mNIS) = \frac{1}{m} k = 1 \sum q [(e_{j k}^{1} - (d_{k}^{1})^{-})^{2} + (e_{j k}^{2} - (d_{k}^{2})^{-})^{2} + \dots + (e_{j k}^{m} - (d_{k}^{m})^{-})^{2}] .$
6. Compute the relative m-PF linguistic closeness coefficient of each alternative a_j by Eq. (5): E'_j = D_e(a_j, mNIS) / (D_e(a_j, mPIS) + D_e(a_j, mNIS)) ∈ [0,1]. Rank the alternatives in descending order of E'_j; the alternative with the largest E'_j is the best. The source gives no tie rule (its case study has no ties); equal E'_j keep the input order.
$E_{j}^{'} = \frac{D _{e} ( a _{j} , mNIS )}{D _{e} ( a _{j} , mPIS ) + D _{e} ( a _{j} , mNIS )} \in [0, 1], j = 1, 2, \dots, p; a_{j_{1}} ⪰ a_{j_{2}} ⪰ \dots ⪰ a_{j_{p}} ⟺ E_{j_{1}}^{'} \geq E_{j_{2}}^{'} \geq \dots \geq E_{j_{p}}^{'} .$

Sources

Adeel, A., Akram, M., & Koam, A. N. A. (2019). Group Decision-Making Based on m-Polar Fuzzy Linguistic TOPSIS Method. Symmetry, 11(6), 735. DOI: 10.3390/sym11060735

Chen, J., Li, S., Ma, S., & Wang, X. (2014). m-Polar fuzzy sets: An extension of bipolar fuzzy sets. The Scientific World Journal, 2014, 416530. DOI: 10.1155/2014/416530

Hwang, C. L., & Yoon, K. (1981). Multiple Attribute Decision Making: Methods and Applications — A State-of-the-Art Survey. Springer-Verlag. DOI: 10.1007/978-3-642-48318-9

Zadeh, L. A. (1975). The concept of a linguistic variable and its application to approximate reasoning—I. Information Sciences, 8(3), 199–249. DOI: 10.1016/0020-0255(75)90036-5