Proximity-Adjusted WLC

This is a ranking method for when you already hold alternatives with known locations (plots of land, homes, facility sites) and a reference point (a city centre, a service point, the decision-maker's own location). Its output is a fit score for every alternative and a descending order based on that score. What sets it apart from classical weighted linear combination (WLC) is that the criterion weights are not the same for every alternative; each alternative is assessed with its own set of weights, according to its distance from the reference point. It was proposed by Rinner and Heppleston in 2006 to show how geographic location affects decision weights in a home buyer's housing-selection problem; it is treated systematically in Malczewski and Rinner's (2015) book on geographic-information-system-based multi-criteria analysis.

The idea behind classical WLC is that every criterion enters the decision with a fixed weight: a criterion's importance does not depend on which alternative is being considered. In spatial decisions this assumption does not always hold. For a plot close to a city centre, "proximity to public transport" is largely an advantage already secured, and other criteria may come to the fore; for a plot far from the centre, the same criterion can be far more decisive. The method's idea is to treat criterion weight not as a fixed number but as a variable that depends on location.

The philosophical consequence is that the method preserves classical WLC's total weight: summed across all alternatives, every criterion's weight still equals its original value, only redistributed among the alternatives according to their proximity to the reference point. Alternatives close to the reference point take the larger share of the weight; distant alternatives take a smaller share. This turns the intuition "location affects the decision" into a mathematical redistribution rule; the method remains compensatory, because it still produces a single weighted total score in the end.

The method proceeds through five steps.

First, scale equalisation. Every criterion is scaled to the range [0, 1]: for a "higher is better" criterion, the value is divided by the largest value; for a "lower is better" criterion, the smallest value is divided by it. If the data is already given in the range [0, 1], this step is skipped.

Second, standardised distance. The straight-line (Euclidean) distance from every alternative to the reference point is calculated. This distance is then divided by the smallest distance among all the alternatives and inverted: the alternative or alternatives closest to the reference point take a value of 1.0, and more distant alternatives take a value below 1.

Third, distributing weight by location. Every criterion's fixed weight is redistributed among the alternatives in proportion to the standardised distance. An alternative close to the reference point receives a higher effective weight on that criterion, and a distant alternative receives a lower one. Summed across all alternatives, the total of these weights remains equal to the number of alternatives multiplied by the original weight; no weight is lost or created.

Fourth, scoring. Every alternative's scaled criterion values are multiplied by its own location-adjusted weights and summed. This is where the difference from classical WLC becomes visible: the same criterion value can be multiplied by a different weight for different alternatives.

Fifth, ranking. Alternatives are ranked from largest to smallest according to this score.

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

The score shows how well an alternative stands once both its criterion values and its proximity to the reference point are taken into account together. This score can differ sharply from the classical WLC score; the same criterion table can produce a very different ranking under a different spatial arrangement. The score itself is not a percentage and cannot be compared with a score from a different analysis, because the choice of reference point directly determines the analysis's result.

An alternative furthest from the reference point can receive a low score even with good criterion values; the reason is not that "it is bad on that criterion" but that its effective weight on that criterion has shrunk. This means the result stems as much from location as from criterion performance, and the report must distinguish between the two.

Thus instead of writing:

"This alternative is the most suitable housing candidate"

the report should read:

"This alternative received the highest spatial fit score relative to the reference point chosen; the ranking could change if the reference point changes"

The method works with crisp data: one number per cell, plus each alternative's geographic coordinates and a reference-point coordinate. You need alternatives in rows, criteria in columns, a positive number in every cell; direction information for every criterion; criterion weights summing to 1; coordinates for every alternative and for the reference point; and a distance measure (the classical application uses straight-line distance). DecisionMind currently holds no extension of this method; it works in its base, crisp form.

The suggested number of alternatives is between 5 and 100, and the number of criteria between 2 and 8; this is not a requirement but the range in which the method works most comfortably. A minimum of two alternatives and one criterion is required.

This method is suitable if your alternatives can be located geographically and you believe that proximity to a reference point should meaningfully affect the decision. Its typical fields are housing-quality assessment, facility-site selection, urban planning, and geographic-information-system-based suitability analysis.

There are three situations where it should not be used. If there is no meaningful spatial relationship among the alternatives (if, say, all alternatives are different floors of the same building and so location-independent), the method's basic assumption is void. If straight-line distance is not realistic (mountainous terrain, an uneven road network, physical barriers), the result becomes misleading. If the choice of reference point is contested, or there is no single correct reference point, the result becomes a reflection of the dispute over the reference point rather than of the decision itself.

Geographic location affects the decision, the reference point can be justified → Proximity-Adjusted WLC

Geographic location has no effect, or a negligible one → classical WLC or TOPSIS

Straight-line distance is not realistic (uneven terrain, road network) → an adaptation working with real road/network distance is needed; in DecisionMind this is a separate engine's job

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

The method's greatest strength is that it explicitly models a fact often overlooked in spatial multi-criteria decisions: location changes the relative importance of criteria. Doing so while preserving the total weight makes the method a natural, traceable extension of classical WLC; the same total weight is merely redistributed among the alternatives. The computational burden is light, and it can be implemented directly within geographic-information-system software.

The method's most notable limitation is that, by definition, it carries rank instability: when a new alternative is added or removed, the sum of standardised distances across all alternatives changes, and this redistributes the weights and scores of the existing alternatives (Ligmann-Zielinska and Jankowski, 2012). This is not a "fault" in the classical sense, since it arises from the method's own spatial-redistribution logic, but it can be misleading if not reported. Second, the result is highly sensitive to the choice of reference point; shifting the reference point by a small distance can reverse the rankings. Third, the standardised distance's value of "1.0" is not assigned to a fixed location but to whichever alternative is currently closest; when the closest alternative changes, the whole scale is recalculated. Fourth, the method assumes straight-line distance; in the real world, travel time or road distance can give a different ranking.

The most common mistake is choosing the reference point without justification or at random; the reference point is part of the decision, and its choice must be defended. A second mistake is adding a new alternative once the analysis is finished and being surprised that the existing alternatives' scores change; this follows from the method's nature and is not an error, but it should be expected in advance. A third mistake is using straight-line distance unquestioningly; where terrain is uneven or a road network exists, this measure does not reflect real accessibility. A fourth mistake is reporting the result without confirming that the sum of the adjusted weights is preserved (the number of alternatives multiplied by the original weight, for every criterion); this check is a simple sign that the calculation was set up correctly.

The governing principle is this:

A Proximity-Adjusted WLC result is a consistent summary of the reference point, the distance measure and the alternative set you supplied; if the reference point or the alternative set changes, the ranking changes too, and the report must show this plainly.