Methods · Objective weighting
WENSLO (Weight by Envelopment and Slope)
An objective method that treats each criterion's cumulative values as a zigzag line and derives criterion weight from the ratio of that line's length to its average slope.
Base method's data type: Classical
What Is the Method?
WENSLO is not a ranking method; it does not rank alternatives, it generates weights for criteria. Given a numerical decision table of alternatives and criteria, the method likens each criterion's column to a geometric shape and assigns weight according to how "long and steep" that shape turns out to be. These weights then feed into a ranking method such as TOPSIS, VIKOR or SAW.
The method was proposed by Pamucar, Ecer, Gligorić, Gligorić and Deveci in 2024 alongside a ranking method called ALWAS; the two together were applied to assessing green-growth performance. WENSLO carries a geometric logic related to the SPC method proposed earlier by the same team.
The Philosophy Behind It
The idea behind WENSLO is an analogy borrowed from engineering drawing. The values in a criterion's column are treated as a cumulative line built by adding one value after another; this line zigzags as it goes, because every alternative carries a value slightly different from the one before. The same line also has an "average slope": a straight reference line advancing at regular intervals, however many alternatives make up the column. WENSLO takes the ratio of the actual zigzag line's length to this straight reference line's slope. The longer and steeper the zigzag, the richer the information the criterion is taken to carry, and the higher its weight comes out.
The consequence of this philosophy is that the weight depends not only on how spread out the values are, but also on the order in which they arrive one after another. This is a fundamental difference from Entropy, CRITIC or SD-WEIGHT; in those methods the order of the alternatives does not affect the weight, whereas in WENSLO it does. If this trait is acceptable, for instance where the alternatives are listed in chronological order or some other natural sequence, WENSLO is the right choice; if the order of the alternatives is arbitrary, this fact must be stated plainly in the report.
How It Works
The method proceeds through six steps.
First, dividing by the column total. Every criterion's column is divided by its own total, turning each column into a set of shares that sum to 1. This step does not read criterion direction; the distinction between cost and benefit plays no part in it.
Second, the class interval. The gap between the largest and smallest value in the column of shares is scaled according to Sturges' rule, the rule used in statistics for determining the number of classes. This interval shrinks as the number of alternatives grows.
Third, the criterion's slope. Since the column's total is already 1, a hypothetical right-angled triangle is constructed: one side is (the number of alternatives minus one) multiplied by the class interval, and the other side is 1. This triangle's slope is taken as the criterion's "average slope."
Fourth, the envelope of cumulation. The actual difference between successive shares in the column, together with the class interval, is treated as the segments of a zigzag line, and the total length of these segments (as a straight-line distance) is computed. This is the criterion's "envelope length."
Fifth, the envelope-to-slope ratio. The envelope length is divided by the average slope found in the third step. A long envelope divided by a small slope signals that the criterion carries rich variability.
Sixth, normalisation. The criteria's envelope-to-slope ratios are summed, and each is divided by that total, converting them into a weight vector that sums to 1.
The formulas behind each step and the intermediate tables are given on the DecisionMind method page; this card carries no formulas.
How to Read the Output
A WENSLO weight does not show a criterion's importance in the decision-maker's eyes; it shows how long and variable a zigzag the criterion's column geometrically traces. A high weight does not mean "this criterion is the most important"; it means "this criterion's cumulative variation across alternatives is the richest relative to its average slope." WENSLO also operates without reading criterion direction; the distinction between "good" and "bad" plays no part in the weight, and enters only at the subsequent ranking step.
WENSLO carries a reading caution of its own: the weight depends on the order in which the alternatives are listed in the table. If the same set of alternatives is entered in a different order, the zigzag line changes and the weights can change with it. WENSLO weights are therefore meaningful only for the alternative order used in that table; it is wrong to change the order and expect the same weights.
Thus instead of writing:
"The WENSLO analysis proved the first criterion is the most important criterion"
the report should read:
"With this alternative order, the first criterion's cumulative variation comes out richest; the WENSLO weight reflects this, not the decision-maker's priority order, and this weight also depends on the order in which the alternatives are listed"
Data Type and Inputs
WENSLO works with crisp data and requires a positive column total. DecisionMind does not hold a separate extension member of WENSLO within its own (objective-weight) family. A fuzzy sibling method that carries the same geometric envelope-slope idea, B-WENSLO, is a separate method; because B-WENSLO is derived not from raw data but from experts' linguistic scores, it sits in the subjective-weight family in DecisionMind.
You need alternatives in rows, criteria in columns, one number per cell, no empty cells, and the order in which alternatives are entered into the table must be meaningful or at least fixed. Weights are not entered; the method produces them. A minimum of two alternatives and two criteria is required, and three to twelve criteria work comfortably. If the alternative order is random or alphabetical, this must be stated plainly in the report, because WENSLO's result is sensitive to that order.
When to Use It, When Not To
WENSLO is a sound choice if you want weight to be quick and data-derived, and the order in which alternatives are entered into the table is not arbitrary, for instance where a chronological, geographic or other natural order applies. It has been used in applications close to the context of its founding paper, such as green growth and environmental performance.
The situations where it should not be used follow from its philosophy. WENSLO is unsuited to a case where the alternative order is entirely arbitrary and this is not acceptable, because the weight comes out dependent on the order; in that case, an order-insensitive method (Entropy, SD-WEIGHT, CRITIC, SPC) should be preferred. WENSLO is also unsuited if you want criterion direction to be reflected in the weight, because the method does not read direction at all.
Alternative order is meaningful or fixed, weight should come from the data quickly → WENSLO
Alternative order is arbitrary and the weight should not be affected by it → Entropy, SD-WEIGHT, CRITIC, SPC
The data consists of experts' linguistic scores → B-WENSLO (subjective-weight family)
The relationship between criteria should also enter the calculation → CRITIC
The decision-maker's priority should show up in the result → AHP, BWM, SWARA (subjective)
Strengths
WENSLO's most important strength is that it measures dispersion not only by the magnitude of the values but also by the order in which they occur one after another; where there is an order or continuity among alternatives (a time series, a gradual progression), this captures information other methods may miss. The geometric analogy the founding paper offers (a zigzag line, a slope) makes the result explainable visually. Its computation is defined in closed form and asks for no additional parameter.
Weaknesses
Its limitations follow from the same design. First and most importantly, the weights depend on the order of the alternatives; the same set of alternatives entered in a different order produces a different weight, a trait confirmed in DecisionMind's own analysis. Second, just like SPC, it does not read criterion direction; direction information only enters at the subsequent ranking step, and a user unaware of this distinction can misread the weight. Third, the method was proposed in 2024 and the number of independent applications is still small; its long-term behaviour has not been tested as broadly as Entropy's or CRITIC's. Fourth, the class-interval calculation, which rests on Sturges' rule, can produce a crude approximation with a very small number of alternatives (two or three).
Common Mistakes
The most common mistake is reporting a WENSLO weight as "importance." The sentence "the first criterion is the most important" is wrong; the correct statement is that the first criterion's cumulative variation comes out richest for this order of alternatives.
A second mistake is entering the alternatives into the table in a random order, running WENSLO, and then reporting the weights as a fixed fact; the weight changes when the order changes, and this sensitivity must be reported. A third mistake is assuming that the WENSLO weight also reflects criterion direction; WENSLO never reads direction. A fourth is placing excessive trust in the weights produced from a table with very few alternatives (two or three); Sturges' rule becomes crude at this scale. A fifth is carrying the weights from this table over to a different alternative set or a different alternative order.
The governing principle is this:
A WENSLO weight measures the richness of a criterion's cumulative variation for this particular order of alternatives; the weight changes when the order changes, and the report must state this dependency plainly.
Cases
Each case opens with a decision table, describes in words what the method does to it, and shows how to read the resulting weights. The first case is DecisionMind's validation example; the figures have been recomputed and verified in Python using the steps the founding paper defines. The remaining cases are illustrative constructions.
1. Illustrative example: Three alternatives, three criteria (DecisionMind validation example)
This example is not taken from the founding paper's own case study; it is a small synthetic table built to make the method's steps traceable by hand, and it is used to validate DecisionMind's WENSLO engine. Three alternatives are assessed on three criteria; the column totals have been chosen to be round numbers (ten each).
| Alternative | K1 | K2 | K3 |
|---|---|---|---|
| A1 | 2 | 3 | 5 |
| A2 | 3 | 4 | 3 |
| A3 | 5 | 3 | 2 |
The method first divides each column by its own total (ten); the K1 and K3 columns carry a symmetric range (from 0.2 to 0.5, a gap of 0.3), while K2 sits in a narrower range (from 0.3 to 0.4, a gap of 0.1). The class interval, slope, zigzag envelope and envelope-to-slope ratio are then computed.
| Criterion | Envelope-to-slope ratio | Weight |
|---|---|---|
| K1 | 0.0892 | 0.4575 |
| K2 | 0.0166 | 0.0851 |
| K3 | 0.0892 | 0.4575 |
The result reads as follows. Although the K1 and K3 columns consist of different numbers (2-3-5 and 5-3-2), both carry the same wide range (0.3), so they receive the same envelope-to-slope ratio and the same weight. K2's range is three times narrower, so its envelope is shorter too, and its weight comes out markedly lower. Here WENSLO has looked only at the width of the range for K1 and K3, not at which alternative is high and which is low on each.
The analyst's hesitation is this: what would happen if the alternatives were entered in the order A3, A2, A1? The columns' totals and ranges would not change, but the difference between successive shares (the zigzag segments) would form in a different order, and the envelope length could change. A team using WENSLO should therefore fix the alternative order before the analysis and state that order in the report.
In the report: "The weights were derived using WENSLO's envelope-to-slope ratio, with the alternatives in the order A1-A2-A3; K1 and K3's equal, high weight (0.4575) arises because these two criteria's range is three times as wide as K2's."
Source: constructed according to the steps defined in Pamucar, Ecer, Gligorić, Gligorić, and Deveci (2024); this is a DecisionMind validation example, not a case taken verbatim from the book or the paper.
2. Logistics: Weighting a courier company's distribution-centre performance
A courier company will weight the performance measures of four distribution centres, listed in the order they opened: daily parcel volume, on-time delivery rate, damage rate, and number of customer complaints. The centres are ordered by opening date, and this order is meaningful to the company, because later centres were set up drawing on the experience of the earlier ones.
The method divides the four columns by their totals and computes each criterion's envelope-to-slope ratio. Suppose the on-time delivery rate shows a steady improvement across the centres in order of opening (a learning effect), forming a long zigzag envelope and receiving a high weight; the damage rate, by contrast, shows irregular ups and downs across the centres, giving a different envelope length.
The company's hesitation: these weights rest on the opening order. If the centres were reordered by geographic size, the zigzag lines would trace a different path and the weights could change. The company has noted that using WENSLO without deciding which order is the "correct" order for the analysis would undermine the report's reliability.
In the report: "The performance weights were derived with WENSLO according to the centres' opening order; the high weight on the on-time delivery rate reflects the steady improvement driven by the learning effect across centres. The weights depend on this opening order."
3. Public sector: Weighting a fire department's station equipment needs
A fire department will weight the equipment-renewal criteria of five stations, listed by year of establishment: average vehicle age, annual number of call-outs, equipment score per staff member, and number of months since the last maintenance. The department wanted the weights to be derived from field data rather than a central commission's opinion.
The method divides the four columns by their totals and computes the envelope-to-slope ratios. Suppose the annual number of call-outs receives the highest weight because it varies irregularly and over a wide range across the stations; the equipment score per staff member receives a lower weight because it stays within a similar range across the stations.
The department's hesitation: the stations could also have been ordered by geographic region rather than by year of establishment. The department has decided to rerun WENSLO with two different orders and check how much the weights change; if the sensitivity to order is large, reporting the result with only one order would be misleading.
In the report: "The equipment-need weights were derived with WENSLO according to the stations' year-of-establishment order; the high weight on the number of call-outs reflects the irregular difference between stations. The weights' sensitivity to station order has been checked separately."
4. What Not to Do
In the illustrative example, it would be wrong to expect the same weights if the alternatives were entered in a different order, such as A2-A1-A3; a WENSLO weight depends on the alternative order. A second error is reporting K1 and K3's equal weight of 0.4575 as "these two criteria are equally important in the decision-maker's eyes too"; the equality holds only for the envelope length in this particular set of three alternatives. A third error is, after computing the WENSLO weights, forgetting which criterion runs in the "good" direction and which in the "bad" direction and feeding the weights directly into ranking; direction information must be added separately after WENSLO.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/wenslo
Pamucar, D., Ecer, F., Gligorić, Z., Gligorić, M., & Deveci, M. (2024). A novel WENSLO and ALWAS multicriteria methodology and its application to green growth performance evaluation. IEEE Transactions on Engineering Management, 71, 9510–9525. DOI: 10.1109/TEM.2023.3321697
Diakoulaki, D., Mavrotas, G., & Papayannakis, L. (1995). Determining objective weights in multiple criteria problems: The CRITIC method. Computers & Operations Research, 22(7), 763–770. DOI: 10.1016/0305-0548(94)00059-h
Wang, Y.-M., & Luo, Y. (2010). Integration of correlations with standard deviations for determining attribute weights in multiple attribute decision making. Mathematical and Computer Modelling, 51(1–2), 1–12. DOI: 10.1016/j.mcm.2009.07.016
Odu, G. O. (2019). Weighting methods for multi-criteria decision making technique. Journal of Applied Sciences and Environmental Management, 23(8), 1449. DOI: 10.4314/jasem.v23i8.7