LMAW (Logarithm Methodology of Additive Weights)

LMAW is a ranking method for when you have a decision table filled with numbers and want the alternatives placed in a single order. Its output is a total utility score for every alternative and a rank built on that score. Pamučar, Žižović, Biswas and Božanić proposed it in 2021, with an application to logistics service provider selection. It offers an alternative to classical weighted-sum methods such as SAW and WASPAS: it first moves cell values onto a logarithmic scale, then passes them through a bounded aggregation function (in the 0–2 range). Like SAW and TOPSIS, it does not produce weights; it takes them from outside. Being a recent method, its literature is still limited. Its main areas of application are supplier and logistics service provider selection.

The idea behind LMAW is to keep the logic of a weighted sum, SAW's philosophy, while limiting how far extreme values in the raw data can drag the ranking. In a classical weighted sum, a very large number on one criterion can dominate the result even after normalisation. LMAW first standardises every cell according to direction information, taking the best or worst value as the reference depending on the criterion's direction. It then moves this standardised value onto a logarithmic scale. A logarithm shrinks large differences while leaving small differences comparatively more visible, compressing the data. In the final step these compressed values are passed, together with the weight, through a function bounded between 0 and 2 and summed. This bound stops a single criterion from inflating the total score without limit.

A consequence follows: LMAW, like TOPSIS and SAW, is compensatory. A weakness on one criterion can be offset by strength on another, because every criterion's contribution is combined into a single total score. What sets LMAW apart from SAW is how this compensation happens. In SAW contributions are directly proportional. In LMAW, because of the logarithmic compression and bounded summation, a very large advantage translates into a proportionally smaller contribution. Where no compromise on a criterion can ever be accepted, that is, where a non-compensatory logic is needed, LMAW's philosophy will not match the decision's own.

The method proceeds through four steps.

First, direction-sensitive standardisation. Every cell is standardised differently depending on the criterion's direction. In a "higher is better" criterion, values are scaled against the largest value; in a "lower is better" criterion, against the smallest. This step is a form of standardisation distinct from TOPSIS's vector normalisation and is specific to LMAW.

Second, the logarithmic transformation. Every standardised value is converted into a logarithm normalised against the product of the values in its own column. This step shrinks large differences and moves the data onto a common, bounded scale, preventing extreme values from gaining disproportionate weight in the next step.

Third, weighted aggregation. The logarithmic value is passed, together with the criterion's weight, through a function bounded between 0 and 2. This function amplifies the contribution of a heavily weighted criterion while stopping a single cell from inflating the total score without limit.

Fourth, summation and ranking. The compressed values across all criteria for each alternative are summed to give a single total utility score. Alternatives are ranked from the highest score to the lowest.

The formulas behind each step, the intermediate tables and citation formats are given on the DecisionMind method page; this card carries no formulas.

The total utility score tells you how high a weighted, logarithmically aggregated total an alternative achieved relative to the others in this particular analysis, and nothing more. The score itself is not a directly interpretable proportion, as it is in SAW; it is a scale specific to LMAW. It does not sit between 0 and 1, and it cannot be compared with scores from a different analysis, because standardisation is built anew from each analysis's own alternative set. The ranking can change if the alternative set or the weights change.

Thus instead of writing:

"LMAW found the best supplier"

the report should read:

"With these weights and this alternative set, the alternative with the highest total utility is this one; the ranking is sensitive to the weight on these criteria"

LMAW works with crisp data: a single number per cell, accepting neither zero nor negative values, because the logarithmic transformation requires a positive value. You need alternatives in rows, criteria in columns, a positive number in every cell with no empty cells; direction information for every criterion; and criterion weights summing to 1. LMAW does not produce weights, it requires them; you can derive them from expert opinion (AHP, BWM, SWARA) or from the data itself (Entropy, CRITIC, MEREC, DEMATEL). DecisionMind holds three LMAW members alongside the base method, including fuzzy extensions. A minimum of two alternatives and two criteria is required; three to twelve criteria work comfortably.

LMAW is a suitable choice where your criteria can be measured numerically, your table has no gaps, you accept that a weakness on one criterion may be offset by strength on another, and your data carries some very large extreme values on certain criteria. If you do not want these extreme values to drag the ranking to excess, LMAW limits this. Its typical fields are supplier and logistics service provider selection; its logic does not depend on the field.

It should not be used in the following situations. Where no compromise is acceptable on a criterion, LMAW is unsuitable, because it is compensatory like TOPSIS and SAW and carries the same limitation. Where your data contains zero or negative values it is also unsuitable, because the logarithmic transformation cannot accept this; the data must first be transformed. Where criteria are strongly linked, this needs handling first.

Numerical table, compensation accepted, you want to limit the effect of extreme values → LMAW

No problem with extreme values, a simple weighted sum will do → SAW, WASPAS

No compromise allowed on one criterion, elimination logic is needed → the ELECTRE family

Not a ranking but weights are needed → AHP, BWM, SWARA (subjective); Entropy, CRITIC, MEREC (objective)

Data contains zero/negative values → transform the data first, or choose a method that does not require a logarithm (TOPSIS, SAW)

LMAW's most important strength is that it limits, through logarithmic compression, how far extreme values in the raw data can drag the ranking to excess. This is particularly useful in data sets where criteria carry a large difference in magnitude, for instance where one criterion sits at an order of 10 while another sits at an order of 10,000. The computational burden is small and the steps can be followed on the table. The weighted aggregation function keeps the result within a bounded range, making it harder than in SAW for a single criterion to dominate the total score disproportionately.

Its limitations stem from this same structure. First, because the method was proposed in 2021, its literature is still limited; it lacks the breadth of independent scrutiny built up around rank reversal and sensitivity to criterion count that TOPSIS or the ELECTRE family enjoy. Second, the logarithmic transformation does not accept zero or negative values; a data set containing these must first be transformed. Third, the assumption of full compensation is the same as in SAW and TOPSIS: a serious weakness on one criterion can be papered over by others. Fourth, criteria are treated as independent; where criteria influence one another, weight can be implicitly counted twice (Božanić, Pamučar, Milić, Marinković and Komazec, 2022).

The most common mistake is applying LMAW without noticing a zero or negative value in the data set; the logarithmic transformation is then undefined. A second mistake is marking criterion direction wrongly. The standardisation step works differently by direction: "higher is better" is scaled against the largest value, "lower is better" against the smallest. A direction error therefore corrupts the ranking directly. A third mistake is assigning equal weights without justification; this is itself a claim that must be defended in the report. A fourth is comparing LMAW's score directly with a score produced by SAW or TOPSIS; each method's scale is its own. A fifth is expecting two methods to give the same ranking on the same data set without recognising that LMAW reflects an extreme advantage on a single criterion more mildly than SAW does.

The governing principle is this:

An LMAW result is a logarithmically compressed summary of the directions, weights and alternative set you chose. If you want to limit the effect of extreme values in the data, this is an advantage. But if one of the inputs is contested, the ranking stays contested too.