Extension card · Stochastic
D-number TOPSIS (Fei, Hu, Xiao, Chen & Deng, 2016)
This is the form of TOPSIS for situations where a criterion value is given not by complete, exact evidence but by a body of evidence that can remain incomplete or partial. It reduces every cell from this evidence to an expected value, and processes the rest exactly as crisp TOPSIS does.
Base method
TOPSIS →
Philosophy, mechanics, strengths and weaknesses are on the base method card; this card describes only the difference.
Data type (family)
Stochastic →
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?
One thing is added; the rest is crisp TOPSIS itself.
Cells. In crisp TOPSIS every cell is a single number. In D-number TOPSIS every cell consists of one or more (value, evidence mass) pairs: for example, an expert can give an alternative's value on a criterion as "4 points, and I assign this assessment a mass of 0.9," leaving the remaining 0.1 unassigned to any value, left open. The sum of the evidence masses cannot exceed 1, but it need not be completed to 1 either; this is the first point that separates a D-number from a probability distribution.
How a D-number differs from a probability distribution and a Z-number. In a probability distribution (this family's founder is covered on the Stochastic data-type card), the sum of all probabilities is always exactly 1; even a state of "I know nothing" is expressed by spreading it across a distribution, such as a uniform one. In a D-number, by contrast, the evidence mass can stay incomplete, and this incompleteness is not shared out elsewhere; a state of "this much I know, I am not speaking about the rest" is represented directly. A Z-number, meanwhile, answers a different question: in the pair Z=(A,B), A is the value itself, and B is the reliability of the source giving that value; the two components are always present together. A D-number has no such second "reliability" component; the evidence mass itself can simply remain incomplete. As an example: "delivery time is 14 days, and I assign 90 per cent evidence to this" is a D-number; the remaining 10 per cent is attached to no value at all. "Delivery time is approximately 14 days (A), and this estimate comes from a reliable source (B)" is a Z-number. "Delivery time, from the distribution of the past 24 months, averages 14 days with a deviation of 3 days" is a probability distribution.
Reduction to an expected value. This is the one step D-number TOPSIS adds: every cell's D-number is converted to a single crisp number by taking the weighted average of its values against their evidence masses (E(D) = Σ mass × value). From this step onward, the resulting matrix is a crisp decision table.
Everything else is crisp TOPSIS. On this crisp table, vector normalisation, weighting, building the ideal and anti-ideal points, Euclidean distance and the closeness score are all applied exactly as in crisp TOPSIS, with no change whatsoever.
DecisionMind fixes, for this extension, the reduction to an expected value and the crisp vector normalisation that follows it. Where more than one expert has given a separate D-number, these can be combined into a single D-number with Dempster's rule of evidence combination before the reduction to an expected value; with a single expert this step is skipped.
How to Read the Output
The closeness score is read as in crisp TOPSIS: it states how close an alternative sits to the ideal, for this alternative set and these weights. The difference lies here: the number in the cell is not a measurement but an expected value derived from partial evidence. The further the evidence mass falls short of 1, that is, the more incomplete it is, the less this expected value rests on. But this incompleteness does not show up in the closeness score, because the moment a cell is reduced to a single number, the information about how complete the evidence was is lost.
Thus instead of writing:
"According to D-number TOPSIS, this alternative is the best"
the report should read:
"With the expected values derived from the evidence masses given, this alternative holds the highest closeness score; in cells where the evidence mass is incomplete, this score is not as robust as in cells resting on complete evidence"
When to Prefer This over the Base Method
This extension suits situations where an expert or source contributes to an assessment with a partial evidence mass, and where the remaining mass's whereabouts is unknown. The typical situation is one of early-stage decisions where more than one evidence source (expert, sensor, record) gives different, not fully overlapping assessments for the same cell, and where evidence-gathering has not yet been completed. If the evidence mass already completes to 1, that is, nothing is left incomplete, the D-number is equivalent to either a single crisp value or a probability distribution, and this extension adds nothing extra. Crisp TOPSIS's exit condition applies here too: where one criterion cannot be traded away, this extension remains compensatory as well and does not screen out anything below a threshold.
Mistakes Specific to This Extension
Sharing out an incomplete evidence mass somewhere. If the mass total is less than 1 (Σv_k<1), sharing this out to another value or to an average breaks the definition of the D-number; the entire point of a D-number is that whatever remains is not assigned anywhere.
Confusing the D-number with the Z-number. The D-number comes from evidence theory (Deng, 2012) and works with a single component (value, mass); the Z-number carries two separate components, a constraint and a reliability (Zadeh, 2011). The two are different frameworks.
Averaging more than one expert's D-numbers without combining them. Where experts have given separate D-numbers, these should first be combined into a single D-number with Dempster's rule of evidence combination; taking a direct numerical average ignores the structure of the evidence.
Presenting the expected value as an exact measurement. E(D) is an expected value; it does not carry how complete the evidence behind the cell is. This information must be stated separately in the report.
The governing principle is this:
D-number TOPSIS's expected value is only read honestly once the report separately states how incomplete the evidence mass was; sharing out the missing mass somewhere, or confusing the D-number with the Z-number or a probability distribution, conceals the method's core contribution.
Cases
The first case is DecisionMind's verification example: the 3x3 D-number example in Fei et al.'s (2016) article; the figures have been computed independently by this card's author, running the kernel directly. The second case is an illustrative construction.
1. Illustrative example (DecisionMind's verification example): A hiring decision among three candidates
A hiring committee assesses three candidates on three criteria: technical competence and communication skill (higher is better), adaptation period (lower is better, on a scale measured in days). Every cell is a D-number: a value and the evidence mass given to that value; where the mass is below 1, the remainder is assigned to no value.
| Candidate | Technical competence (value; mass) | Communication (value; mass) | Adaptation period (value; mass, lower is better) |
|---|---|---|---|
| K1 | 3; 0.8 | 5; 1.0 | 4; 0.9 |
| K2 | 5; 1.0 | 3; 0.8 | 2; 1.0 |
| K3 | 4; 0.7 | 4; 1.0 | 3; 0.8 |
| Weight | 0.40 | 0.35 | 0.25 |
The method first reduces every cell to an expected value (value × mass): 2.4 / 5.0 / 3.6 for K1; 5.0 / 2.4 / 2.0 for K2; 2.8 / 4.0 / 2.4 for K3. On this crisp table, vector normalisation, weighting, the ideal-anti-ideal point and Euclidean distance are applied exactly as in crisp TOPSIS.
| Candidate | Closeness score | Rank |
|---|---|---|
| K2 | 0.585 | 1 |
| K1 | 0.415 | 2 |
| K3 | 0.412 | 3 |
The result reads as follows. K2 holds the highest expected value on technical competence, the heaviest criterion, and also the shortest adaptation period; K1 and K3 finish very close to one another, with a gap of 0.003.
The committee's hesitation: when the weight is concentrated on communication (0.30; 0.60; 0.10) and recomputed, K1 (0.637) overtakes K2 (0.363); this shows that K2's first place depends on the weight given to technical competence and adaptation period (independently recomputed with the kernel). Furthermore, the 0.003 gap between K1 and K3 is not a robust ranking, because neither candidate's evidence mass is complete (between 0.7 and 0.9); it is difficult to make a firm choice between these two candidates before the mass shortfall is completed.
In the report: "With the evidence masses given, K2 holds the highest closeness score (0.585); when the weight is concentrated on communication (0.30; 0.60; 0.10), K1 moves ahead. The gap between K1 and K3 (0.415 against 0.412) rests on cells with incomplete evidence mass and should not be taken as robust."
Source: Fei, Hu, Xiao, Chen and Deng (2016), Mathematical Problems in Engineering, the article's 3x3 D-number example. This card's author has independently run DecisionMind's engine; the engine's own expected-value calculation (value × mass, with the remaining mass shared nowhere) produces a figure that differs from some of the manifest's internal documentation notes, detailed in the verification notes. The ranking (K2 first) is the same in both calculations.
2. Logistics: Incomplete evidence in choosing a freight carrier
An e-commerce company will contract with one of three freight carriers. Three criteria apply: on-time delivery rate, damage rate (lower is better) and customer-complaint score (lower is better). The company's operations team gives, for every carrier, a value based on its own audit reports and the evidence mass with which it agrees with that value, namely the share of shipments actually audited; for shipments not audited, the evidence mass is left incomplete.
The method reduces every cell to an expected value and applies crisp TOPSIS. Suppose the carrier with the highest on-time delivery rate is also the least-audited carrier; it still comes out ahead on the closeness score, because the calculation looks only at the expected value.
The team's hesitation is this: because this carrier's evidence mass is low, the expected value may diverge from actual performance. The team should look not only at the closeness score but also at which carrier's audit coverage, that is, whose evidence mass, is more complete.
In the report: "Based on the expected values, the first carrier stands out; this carrier's audit coverage is narrower than the others', so an additional audit is recommended before contracting."
3. What Not to Do
In the illustrative example, sharing out K1's incomplete mass on technical competence (0.2) to another value, for instance the highest score, and computing the expected value as 3.4: this breaks the D-number's principle that "whatever remains is assigned nowhere," and produces a fabricated number that differs from the 2.4 the engine actually computes. The second mistake is confusing the D-number with the Z-number and presenting the evidence mass as though it were a "reliability" component; the two are separate frameworks. The third mistake is reporting the 0.003 gap between K1 and K3 as a robust ranking; this gap rests on cells with incomplete evidence mass.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/d-topsis
Fei, L., Hu, Y., Xiao, F., Chen, L., & Deng, Y. (2016). A modified TOPSIS method based on D numbers and its applications in human resources selection. Mathematical Problems in Engineering, 2016, Article 6145196. DOI: 10.1155/2016/6145196
Deng, Y. (2012). D numbers: Theory and applications. Journal of Information & Computational Science, 9(9), 2421–2428. (no DOI)
Zadeh, L. A. (2011). A note on Z-numbers. Information Sciences, 181(14), 2923–2932. DOI: 10.1016/j.ins.2011.02.022
Hwang, C. L., & Yoon, K. (1981). Multiple Attribute Decision Making: Methods and Applications, A State-of-the-Art Survey. Lecture Notes in Economics and Mathematical Systems, Vol. 186. Springer-Verlag. DOI: 10.1007/978-3-642-48318-9