LINMAP (LINear programming technique for Multidimensional Analysis of Preference)

LINMAP is a ranking method for when you have a numerical decision table but would rather make pairwise comparisons, of the form "I prefer this alternative to that one," than state criterion weights directly. Its output is a distance value for every alternative and a rank built on that value; a smaller distance marks the better alternative. Srinivasan and Shocker proposed it in 1973, in marketing and psychometrics, to model consumer preferences; over the following fifty years it moved into the multi-criteria decision-making literature and acquired numerous fuzzy and uncertain extensions.

Most ranking methods ask the decision-maker for weights first and then compute with them. LINMAP reverses this. It does not ask for weights; instead it asks which of certain pairs of alternatives the decision-maker prefers. It then runs a search behind the scenes: which weights and which "ideal alternative", if assumed, would make the ranking by distance to that ideal agree as closely as possible with the preferences the decision-maker has stated. Neither the weights nor the ideal point are asked for directly; both are inferred backwards from the preferences.

This treats preference as data. The decision-maker need not explain why one alternative is preferred to another; they simply state the preference, and the method finds the numerical model that best explains it. Where a preference does not fit (contradictory or cyclical preferences, for instance), the method does not ignore this; it reports how much inconsistency remains through a single figure, the residual deviation.

A philosophical consequence follows: LINMAP, like TOPSIS, is compensatory. Because the distance to the ideal point is computed through a weighted sum, a weakness on one criterion can be masked by strength on another. What sets it apart is that these weights are not supplied from outside; they are derived from the preferences themselves.

The method proceeds through two steps.

First, deriving the weights and the ideal point from the preferences. The decision-maker states a preference for certain pairs of alternatives (alternative A over alternative B, say). The method searches for a set of weights and an ideal point that conflict as little as possible with these preferences. A mathematical device is needed here: because the quantities being sought, the weights and the ideal point, would make the search non-linear if multiplied together directly, LINMAP instead treats the product of a weight and the corresponding ideal-point coordinate as a single new variable, turning the whole search into an ordinary linear programming problem. Weights are constrained to sum to 1 and to be non-negative. How far each preference is violated is tracked by its own "deviation" variable, and the total deviation is minimised.

Second, computing the distances and ranking. Once the linear programme is solved, the actual ideal point and weights are recovered. Each alternative's weighted distance to this ideal point is calculated, and the alternatives are ranked from smallest to largest distance; the alternative closest to the ideal is the best.

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

The distance value shows how close an alternative sits to the ideal point derived from the stated preferences; it is not a percentage or a direct "quality score". This ideal point has been computed solely for the given pairs of preferences; a different set of preferences could produce an entirely different ideal point and a different ranking.

A total deviation of zero means that every preference supplied is fully consistent with the model found. A deviation greater than zero shows that some part of the preferences conflicts with the rest, and that the model has sought the best possible fit by sacrificing some of them; this figure must always be stated in the report, or the reader will assume every preference has been satisfied. Even a solution ending in zero deviation may not be unique: another solution, equally consistent with the same set of preferences but with an entirely different weight distribution, may also exist. Presenting the weights LINMAP finds as "the decision-maker's true criterion priorities" is therefore risky; they are only an explanation consistent with the stated preferences, and may not be the only one.

Thus instead of writing:

"LINMAP revealed the decision-maker's criterion weights"

the report should read:

"An ideal point and a set of weights consistent with the stated pairwise preferences have been found; if this set's uniqueness has not been tested, the weights should not be presented as the only interpretation"

LINMAP works with crisp data: a single number per cell. DecisionMind holds no extension of this method; it is offered only in its base form, working solely with crisp numbers.

You need alternatives in rows, criteria in columns, one number per cell; direction information for every criterion (including "lower is better" criteria such as price and time); and, for at least a few pairs of alternatives, the decision-maker's binary judgement of the form "I prefer this one to that one." LINMAP does not take weights from outside; it derives both the weights and the ideal point from these pairwise preferences itself. A minimum of two alternatives and two criteria is required; three to twelve criteria work comfortably. As the number of stated preferences grows, the model is tested more rigorously, but the chance of a contradiction among the preferences also increases.

LINMAP is a suitable choice where the decision-maker struggles to assign numerical weights to criteria but can say "I prefer this one to that one" between alternatives. It is used in fields such as supplier evaluation and alternative selection, where past choices or pilot preferences already exist; pairwise preferences can be drawn naturally from these earlier choices.

LINMAP is not suitable if the preferences supplied are unreliable, conflict with one another, or are too few; an ideal point built on very few preferences can be nearly arbitrary. Where the decision-maker wants to see clear, defensible numbers behind criterion priorities, the weights LINMAP derives are not on their own a sufficient justification, because they may not be unique.

Pairwise preference comes naturally, weighting is difficult → LINMAP

Weights can be obtained directly from an expert → AHP, BWM, SWARA

Weights should be derived from the data → Entropy, CRITIC

No compromise allowed on one criterion, sub-threshold alternatives must be screened out → screening first, then ranking

LINMAP's most important strength is that it asks the decision-maker for a more natural task, pairwise preference, instead of the difficult task of numerical weighting. It has a history spanning more than fifty years and can be solved exactly through linear programming; this means the result is not a heuristic approximation but the exact solution of a well-defined optimisation problem. It does not silently absorb inconsistent preferences; it reports them openly through the residual deviation value. It has been adapted in the literature to fuzzy, intuitionistic fuzzy and interval-valued settings among others (Li, 2008), showing that the underlying idea transfers to different data types.

Its limitations largely concern the uniqueness of the solution. DecisionMind's own validation of this example shows that more than one weight distribution can exist with zero total deviation; one such distribution might assign all the weight to a single criterion while another assigns it to a different one, both equally consistent with the preferences. In that case the phrase "the weight LINMAP found" can be misleading. A second limitation is that the original method works only with crisp data; uncertain or linguistic assessments cannot be handled directly, which is why the literature has developed fuzzy and intuitionistic fuzzy extensions (Li, 2008). Third, LINMAP is compensatory like TOPSIS; a serious weakness on one criterion can be masked by others. Fourth, the fewer preferences supplied, the less constrained the ideal point is, and the more fragile the result becomes.

The most common mistake is ignoring a total deviation greater than zero and reporting the result as though every preference had been satisfied. A second is presenting the weights found as the decision-maker's "true" criterion priorities without testing whether this solution is unique. A third is working with very few pairwise preferences (a single pair, say) and expecting a strong ideal point. A fourth is entering preferences that conflict cyclically with one another and assuming LINMAP will halt with an error; it does not halt, it searches for the best fit and shows the contradiction only through the deviation figure. A fifth is comparing distance values from different analyses; because each analysis's ideal point is derived from its own preferences, they are not comparable.

The governing principle is this:

A LINMAP result is an inference of the ideal point and weights that conflict least with the stated pairwise preferences; if the deviation is not zero, or the solution is not unique, the report must say so plainly.