Local WLC (neighbourhood-sensitive Weighted Linear Combination)

Local WLC is a ranking method for situations where alternatives sit on a map or within a regional structure, and imposing the classical weighted-sum method's (WLC's) single set of weights across the whole region is unrealistic. Its output is a score between 0 and 1 for every alternative and a rank built on that score. Malczewski proposed it in 2011 for GIS-based multi-criteria analysis; it is a local form of the classical weighted linear combination.

A classical weighted sum makes the decision "this criterion matters this much" once, for the whole region. Malczewski showed that this decision carries an implicit assumption: a criterion's importance actually depends on how much it VARIES. If every alternative is nearly the same on a criterion, that criterion fails to distinguish the alternatives, whatever weight it is given; if the same criterion varies widely in another region, it becomes decisive there. This observation is called the range-sensitivity principle (Keeney, 1992): the wider a criterion's range of variation, the greater the weight it should receive.

Local WLC applies this principle to geographic neighbourhoods. The region is divided into neighbourhoods of nearby alternatives, and within each neighbourhood a criterion's LOCAL range of variation is compared against that criterion's GLOBAL range (across the whole region). This ratio pulls the global weight up or down: a criterion that barely varies within a neighbourhood receives a low local weight, one that varies widely receives a high local weight. The result is a weighted sum adapted to each neighbourhood's own internal structure while still derived from the same overall logic.

The philosophical consequence is that Local WLC remains compensatory, but the rate of compensation now varies by region. In classical WLC, which criterion offsets a weakness on another, and by how much, is fixed; in Local WLC, this rate of offsetting is recalculated according to the neighbourhood an alternative sits in.

The method proceeds through five steps.

First, general scale equalisation. Every criterion is scaled to between 0 and 1 across ALL alternatives, without any distinction by neighbourhood. This step is always done globally, or else scores from different neighbourhoods become incomparable.

Second, computing the global and local ranges. For every criterion, the global range across all alternatives and the local range within each neighbourhood are calculated separately.

Third, building the local criterion weights. For each neighbourhood, the criterion's global weight is multiplied by the ratio of that neighbourhood's local range to the global range and rescaled; all the weights within a neighbourhood are then corrected so that they again sum to 1.

Fourth, computing the local score. Each alternative's scaled criterion values are multiplied by its own neighbourhood's local weights and summed. Alternatives in the same neighbourhood share the same set of local weights.

Fifth, ranking. All alternatives, even if they come from different neighbourhoods, are ranked from highest to lowest by the same score. This comparison is valid because the scale equalisation was always done globally.

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

The score shows an alternative's total performance, weighted by the local weights that apply in its own neighbourhood. A high score shows the alternative is strong on the criteria that discriminate within its own neighbourhood; it does not mean the alternative would be equally strong in a different neighbourhood.

Two alternatives' scores are comparable even if they come from different neighbourhoods, because the scale equalisation is done globally. But when the neighbourhood boundaries change, the local weights change too; the same raw data with a different neighbourhood drawing can therefore produce a different ranking. Two alternatives that receive an equal score in classical WLC can diverge in Local WLC because of a neighbourhood difference; this is not an error but the differentiating effect the method is designed to produce, though it must be explained in the report.

Thus instead of writing:

"This alternative is the best alternative"

the report should read:

"This alternative has achieved the highest score under the local weights in its own neighbourhood; the ranking may change if the neighbourhood boundaries change"

Local WLC works with crisp numerical data. DecisionMind holds no extension of this method; it is offered only in its base form, working solely with crisp numbers.

You need a decision table; a grouping showing which neighbourhood each alternative belongs to; direction information for every criterion; and global criterion weights. The neighbourhood grouping is a decision made from outside, by the analyst or by a geographic or administrative structure; the method does not produce this grouping. Every neighbourhood must contain at least two alternatives, or the local range cannot be calculated, and a criterion could be found artificially non-discriminating in that neighbourhood. A recommended size is at least four alternatives and between two and eight criteria; typical fields of application are land-suitability analysis, ecological zoning, urban vulnerability assessment and regional planning.

Local WLC is a suitable choice if your alternatives can be meaningfully grouped into a geographic or structural neighbourhood, and you believe the same criterion discriminates to different degrees across different regions. It has been used in studies such as heat-related health-risk assessment (Carter and Rinner, 2014) and, more broadly, in GIS-based land and regional assessments.

Local WLC is unsuitable if your neighbourhood boundaries are arbitrary or unjustified, or if any neighbourhood has fewer than two alternatives; in that case the boundaries should be redrawn, or classical WLC used with a single fixed weight across the whole region. If you want to express a risk attitude based on the order of criteria (best, second-best), LOCAL-OWA is more suitable than Local WLC.

Criterion importance varies by region, a direct weighted sum is sufficient → Local WLC

Criterion importance varies by region, a risk attitude (optimistic/pessimistic) is also to be expressed → LOCAL-OWA

Criterion importance is the same everywhere → standard WLC

Neighbourhood boundaries unclear or too small → the neighbourhood definition should be reviewed first

Local WLC's most important strength is that it openly questions the classical weighted sum's implicit, and often unrealistic, assumption of "the same weight everywhere". It turns the range-sensitivity principle (Keeney, 1992) into a traceable calculation; which criterion carries more weight in which neighbourhood, and why, can be shown step by step. It can differentiate alternatives that tie in classical WLC by using neighbourhood information, giving a more detailed ranking from the same data. Its adaptation to vector-based geographic information systems (Carter and Rinner, 2014) shows the method is applicable across different data infrastructures.

Its limitations largely concern the choice of neighbourhood. How the neighbourhood boundaries are drawn directly determines the local weights; results can change when boundaries are redrawn, which is why a sensitivity analysis on the neighbourhood scheme is recommended (Malczewski, 2011). A criterion taking the same value for every alternative in a neighbourhood (a local range of zero) is a pathological case; that criterion cannot then be given a local weight, and the method must fall back on the global weight. As the number of alternatives within a neighbourhood shrinks, the estimate of the local range becomes unstable; very small neighbourhoods can produce artificially extreme weights. Finally, Local WLC remains compensatory like classical WLC; a weakness on one criterion can still be offset by another, even if that criterion does not carry a high weight in that neighbourhood.

The most common mistake is performing scale equalisation by neighbourhood rather than globally; this makes scores from different neighbourhoods incomparable. A second mistake is drawing neighbourhood boundaries without reference to the data or to domain knowledge, and not reporting this choice. A third is continuing the calculation without noticing that a criterion's local range has come out at zero in some neighbourhood; this risks division by zero or an artificially nulled weight, and a fallback to the global weight is needed. A fourth is skipping the sensitivity analysis as the number of neighbourhoods grows and presenting a single neighbourhood scheme's result as definitive. A fifth is comparing classical WLC and Local WLC results without explaining which alternatives changed places because of the neighbourhood effect, giving only the final ranking.

The governing principle is this:

A Local WLC result is a sum computed on a global scale but reweighted according to local ranges; the neighbourhood boundaries must be stated clearly in the report, or the reason for the ranking cannot be understood.