Extension card · Fuzzy
Interval-valued intuitionistic fuzzy TOPSIS (Jahanshahloo, Lotfi and Izadikhah, 2006)
The form of TOPSIS for situations where criterion values are given with a lower and an upper bound, that is, as an interval. When only an interval is given, the rejection information is taken as zero, and the result is again ranked by a single closeness coefficient.
Base method
TOPSIS →
Philosophy, mechanics, strengths and weaknesses are on the base method card; this card describes only the difference.
Data type (family)
Fuzzy →
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 carries a support interval and a rejection interval; together the two form an interval-valued intuitionistic fuzzy number. The user most often enters only one interval, for instance giving a duration as "between 3.5 and 4.5 days." In that case the engine takes this interval as the support and treats the rejection interval as zero. This means "no evidence against it," not "full support." If the expert also supplies a separate rejection interval, the same cell format can carry that too. Criterion weights are crisp numbers; the method does not generate weights, it takes them from outside.
Scale equalisation. The support and rejection intervals on cost criteria are swapped first. The whole matrix is then reduced to the 0-1 range by dividing by the single largest value it contains. This differs from crisp TOPSIS, which equalises each column separately according to its own magnitude; here the whole matrix is scaled to one shared magnitude.
Weighting. In crisp TOPSIS, weighting is multiplication. Here the support interval grows as 1−(1−support)^weight, and the rejection interval shrinks as rejection^weight. This applies the weight in a graduated rather than a linear way. As the weight approaches zero, support is driven towards zero; as the weight approaches one, support approaches its raw value. This behaviour means that a small change in weight can affect the result more sharply than in crisp TOPSIS. Case 1 demonstrates this.
Distance and result. The ideal and the anti-ideal are built from the best and worst composite of every column's support and rejection intervals. The distance is a Euclidean distance computed from the differences in the lower support, upper support, lower rejection and upper rejection bounds. The closeness coefficient is again a single number between 0 and 1.
DecisionMind fixes, in this extension, the cost-criterion complement, the rule of dividing the whole matrix by one shared magnitude, and the exponential weighting.
How to Read the Output
The closeness coefficient is read as in crisp TOPSIS, ranking only this alternative set relative to itself. The difference is here: because weighting is not linear, a small change in weight can produce a shift in ranking far larger than would be seen in crisp TOPSIS. If only one interval was entered and rejection was left at zero, the result means "no counter-evidence was sought"; this must not be confused with "there is no counter-evidence."
Thus instead of writing:
"The IV-TOPSIS ranking stays robust even when the weights change slightly"
the report should read:
"Weighting is done through an exponential operation and is not linear; a small change in weight can change the ranking more readily than in crisp TOPSIS, and this sensitivity must be shown in the report"
When to Prefer This over the Base Method
This extension is appropriate when a criterion is known only by a lower and an upper bound, that is, when an expert says "between 3 and 5" rather than a single number. If the expert also gives a separate counter-evidence interval, this too fits the same cell. If the same criterion has several distinct, separately defensible values, that is, a set rather than an interval, the hesitant structure (HF-TOPSIS) should be used instead; here there is a single interval, not a set of separate values. If a criterion is measured, stay with the base method; DecisionMind requires the matrix to hold a single data type. Crisp TOPSIS's compensatory nature and its no-compromise exit condition apply here exactly as they do there.
Mistakes Specific to This Extension
The lower bound exceeding the upper bound. If the interval rule (the lower bound cannot exceed the upper bound) is broken, the column denominator is corrupted and every subsequent calculation becomes invalid.
Confusing the cost-benefit direction. Which bound the ideal point is built from depends on the criterion's direction. If the direction is marked incorrectly, the method builds the ideal from the wrong bound.
Mistaking the interval for a triangular fuzzy number. There is no centre, that is, no most-likely value, here; only a lower and an upper bound exist. Where a three-cornered value genuinely exists, FUZZY-TOPSIS should be used.
Failing to show weight sensitivity. Because weighting is exponential, the ranking can turn on a smaller weight change than in crisp TOPSIS. Case 1 demonstrates this.
The governing principle is this:
The only information in the cell is a lower and an upper bound; if the rejection interval was not separately measured, it is taken as zero, and this assumption must be stated in the report.
Cases
The first case is Jahanshahloo, Lotfi and Izadikhah's (2006) classic 3×3 example. The figures were independently computed with DecisionMind's IV-TOPSIS engine. The second case is an illustrative construction.
1. Manufacturing: Choosing among three suppliers (Jahanshahloo, Lotfi and Izadikhah, 2006)
A manufacturing firm will choose one of three suppliers. Three criteria apply: delivery reliability and product quality are "higher is better," unit cost is "lower is better." Every criterion is given as a lower and an upper bound, that is, an interval; since no separate rejection interval was measured, it is taken as zero.
| Supplier | Delivery reliability | Product quality | Unit cost |
|---|---|---|---|
| A1 | [2.5; 3.5] | [4.5; 5.5] | [3.5; 4.5] |
| A2 | [4.5; 5.5] | [2.5; 3.5] | [1.5; 2.5] |
| A3 | [3.5; 4.5] | [3.5; 4.5] | [2.5; 3.5] |
| Direction | higher is better | higher is better | lower is better |
| Weight | 0.40 | 0.35 | 0.25 |
The method complements the cost column, divides the whole matrix by one shared magnitude, combines the support and rejection intervals exponentially by weight, builds the ideal and anti-ideal supplier, and computes every supplier's closeness coefficient.
| Supplier | Closeness coefficient | Rank |
|---|---|---|
| A2 | 0.502 | 1 |
| A1 | 0.498 | 2 |
| A3 | 0.267 | 3 |
The result reads as follows. A2 has the best intervals on delivery reliability and cost. A1 leads by a clear margin only on product quality. The gap between the two suppliers' total scores is only 0.004; this means the three criteria's weighted total nearly equalises them.
The firm's hesitation is this. If delivery reliability's weight is raised from 0.40 to 0.42 and the other two weights are shrunk proportionately (to 0.338 and 0.242), A1 moves ahead, leaving A2 behind at 0.503 against 0.497. A shift in weight of only about two percentage points reverses the ranking; this stems from the exponential structure of the weighting and would not be seen with so small a shift in crisp TOPSIS.
In the report: "With the given weights, A2 is the supplier closest to the ideal (0.502); the gap to A1 (0.498) is very small. When delivery reliability's weight is raised to 0.42, A1 moves ahead. The choice between these two suppliers depends on the weighting assumption."
Source: Jahanshahloo, Lotfi and Izadikhah (2006), as transmitted in Roszkowska (2011) §3.1. The closeness coefficients were independently computed with DecisionMind's IV-TOPSIS engine. The A1-A3 ranking the engine produces differs from the source's own transmitted ranking; the detail is in the verification note.
2. Logistics: Choosing among three regional distribution centres
A courier company will choose among three candidate sites for a new regional distribution centre. Three criteria apply: daily parcel capacity is "higher is better," and average delivery time and set-up cost are "lower is better." Every criterion is given as a lower and upper estimate from site surveys; no criterion has a single crisp number.
The method computes every centre's interval distance to the ideal and anti-ideal centre and converts it into a closeness coefficient. Suppose the lowest-cost site is also the site with the widest capacity, and it finishes first. The site with the widest delivery-time interval, despite its low cost, finishes last.
The company's hesitation is this. If the upper bound of the site with the widest delivery-time interval is narrowed by an improved site estimate, this site could move up to second place. The interval's width is itself a risk indicator, and looking only at the closeness coefficient conceals this risk.
In the report: "The lowest-cost site has the highest closeness coefficient. The site with the widest delivery-time interval has performance that varies by region; this risk should be separately assessed."
3. What Not to Do
Narrowing A1's delivery-reliability interval in the first case from an interval to [3.0; 3.0], as though the expert had given a single number: this loses the uncertainty information the interval carries. The second error is forgetting that unit cost is "lower is better" and marking it "higher is better" instead; this builds the ideal point from the most expensive supplier and makes the ranking meaningless. The third error is writing the 0.004 gap between A1 and A2 as "A2 is definitively better"; this gap is sensitive to the weighting assumption and has already reversed under a small change in weight.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/iv-topsis
Jahanshahloo, G. R., Lotfi, F. H., & Izadikhah, M. (2006). An algorithmic method to extend TOPSIS for decision-making problems with interval data. Applied Mathematics and Computation, 175(2), 1375–1384. DOI: 10.1016/j.amc.2005.08.048
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
Atanassov, K. T. (1986). Intuitionistic fuzzy sets. Fuzzy Sets and Systems, 20(1), 87–96. DOI: 10.1016/S0165-0114(86)80034-3
Roszkowska, E. (2011). Multi-criteria decision making models by applying the TOPSIS method to crisp and interval data. Multiple Criteria Decision Making '10-11, University of Economics in Katowice, 200–230. (no DOI)