MABAC as a transparent comparison with a criterion-by-criterion border area
The Multi-Attributive Border Approximation Area Comparison method, usually called MABAC, ranks alternatives by comparing each weighted criterion value with a constructed border approximation area. Values above the border contribute positively, values below it contribute negatively, and the alternative’s signed distances are added. The arithmetic is compact; the judgement design is not.
This guide builds MABAC from first principles, works a fictional three-option example, and then tests the ranking under changed weights, ranges, option sets and veto conditions. The central discipline is to keep benefit and cost directions visible, avoid confusing a signed distance with probability, and recognise that a compensatory score cannot repair an invalid criterion or an option that fails a hard constraint.
The worked numbers are invented for learning. A real decision needs an accountable choice of alternatives, criteria, measurements, weights, uncertainty treatment and governance. MABAC is a decision aid, not an automatic decision-maker.
Full chapter index · Independent practice · Judgement subject guide
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Full chapter index
- Frame the decision before applying MABAC
- Build the decision matrix without hiding measurement choices
- Normalise benefit and cost criteria with direction visible
- Work a fictional three-option MABAC example
- Apply weights and construct the border approximation area
- Calculate signed distances and the final score
- Interpret signs without turning them into probabilities
- Test weight, range and option-set stability
- Separate compensation from veto conditions
- Audit criteria, weights and group judgement
- Transfer the method when an extreme option and a veto appear
- Continue through an auditable MABAC pathway
A multi-criteria method begins with a decision statement, not a matrix. Name the decision owner, alternatives, affected people, time horizon and the action that will follow. An option that is unavailable, unlawful or outside budget should not enter the ranking as if it were feasible.
Criteria should represent distinct and decision-relevant concerns. Define each unit, direction and evidence source. A benefit criterion is better when larger; a cost criterion is better when smaller. If direction is ambiguous, the concept is not ready for normalisation.
Separate hard constraints from tradeable preferences. Safety certification, legal eligibility or a non-negotiable capacity floor may be gates. Turning a gate into a lightly weighted criterion can allow a strong score elsewhere to compensate for failure that the decision owner never intended to trade.
Evidence check: Create a decision register with criterion definition, direction, unit, data date, source, uncertainty, weight owner and constraint status. The matrix should be reproducible from that register.
Changed-context transfer: If an alternative becomes unavailable after calculation, remove it and rerun the full method because normalisation and border values depend on the option set.
Limit and misconception check: MABAC cannot determine which values society or an organisation ought to prioritise. It operates on the criteria and weights supplied.
Practical decision record: Write the decision in one sentence, list the feasible options and mark every proposed criterion as gate, benefit or cost before entering numbers.
Let xᵢⱼ denote the observed value for alternative i on criterion j. Rows are alternatives and columns are criteria. Units can differ because normalisation will place them on a common scale, but the underlying values still need comparable definitions.
A score such as service quality or environmental risk must be anchored. State who rated it, the rubric, the period and whether differences are meaningful. A precise decimal from an informal judgement is not made objective by entering it into a formula.
Missing data require an explicit rule. Excluding the option, imputing a value, using an interval or collecting more evidence can each be defensible in a different setting. Silently substituting the column average conceals uncertainty and can distort both the normalised values and the border.
Evidence check: Audit every cell back to a source or a documented elicitation. Inspect outliers and make sure a higher raw number really has the stated direction.
Changed-context transfer: If one supplier reports annual cost and another reports a three-year total, convert them to the same decision horizon before normalisation. The method cannot repair mixed definitions.
Limit and misconception check: Normalisation removes units from calculation; it does not make weak measurements equivalent to strong measurements.
Practical decision record: Select one qualitative criterion and write a rubric with observable anchors for low, medium and high performance. Test whether two reviewers can apply it consistently.
For a benefit criterion, a common min–max transformation is tᵢⱼ = (xᵢⱼ − xⱼmin) divided by (xⱼmax − xⱼmin). The lowest observed value becomes zero and the highest becomes one. For a cost criterion, reverse the direction: tᵢⱼ = (xⱼmax − xᵢⱼ) divided by the same range.
The transformation is option-set dependent. Add a new extreme alternative and the minimum or maximum can change, moving every existing normalised value. That does not necessarily mean an error; it means the reference frame changed.
When all alternatives have the same value, the range is zero and the formula is undefined. The criterion provides no discrimination in that option set. A defensible workflow removes it from ranking, revisits the weight allocation or applies a predeclared convention rather than dividing by zero.
Evidence check: Show raw values and normalised values side by side, with benefit or cost direction in the column heading. Recalculate a few cells manually.
Changed-context transfer: If a cost is transformed into savings relative to a baseline, it may become a benefit criterion. Document the change and verify that the ranking remains logically consistent.
Limit and misconception check: Min–max normalisation is sensitive to extremes and does not encode uncertainty. A difference of 0.1 in the transformed scale is not automatically an important real-world difference.
Practical decision record: Normalise one benefit and one cost column by hand. Then add a dominated extreme option and observe how the old transformed values move.
A fictional team compares three service options. Accuracy is a benefit with weight 0.40, speed is a benefit with weight 0.30, and annual cost is a cost with weight 0.30. Option A scores 80, 6 and 50; Option B scores 70, 9 and 40; Option C scores 90, 7 and 70.
Min–max normalisation gives A the vector 0.50, 0.00 and 0.67; B gives 0.00, 1.00 and 1.00; C gives 1.00, 0.33 and 0.00. The cost column is reversed, so the lowest cost receives one.
The example is intentionally small so every step can be audited. It does not claim that accuracy, speed and cost are sufficient for a real procurement. Safety, capacity, equity or implementation constraints might be gates or additional criteria.
| Alternative | Accuracy benefit w=0.40 | Speed benefit w=0.30 | Cost cost w=0.30 |
|---|---|---|---|
| A | 80 → 0.50 | 6 → 0.00 | 50 → 0.67 |
| B | 70 → 0.00 | 9 → 1.00 | 40 → 1.00 |
| C | 90 → 1.00 | 7 → 0.33 | 70 → 0.00 |
Evidence check: Recompute the cost column from maximum 70 and minimum 40. For Option A, (70 − 50) / 30 is approximately 0.67.
Changed-context transfer: If speed were time-to-complete rather than throughput, smaller would be better and the direction would reverse. A label change can alter the entire ranking.
Limit and misconception check: Rounded values are shown for teaching; use unrounded values through the calculation and round only for presentation.
Practical decision record: Create a fourth option that is worse than B on all three criteria. Predict what should happen, then test whether the option changes the old ranking through rescaling.
MABAC commonly forms weighted normalised values vᵢⱼ = wⱼ(tᵢⱼ + 1). Adding one keeps every value positive when weights are positive. In the example, Option A becomes approximately 0.60, 0.30 and 0.50; B becomes 0.40, 0.60 and 0.60; C becomes 0.80, 0.40 and 0.30.
For each criterion, the border approximation value gⱼ is the geometric mean of the alternatives’ weighted values: multiply the vᵢⱼ values and take the m-th root, where m is the number of alternatives. The example gives approximate borders of 0.577, 0.416 and 0.448.
The geometric mean makes the border depend on every alternative. This is a constructed reference, not a real option, minimum requirement or probability threshold. Its meaning is entirely within the method and current decision matrix.
Evidence check: Verify that weights sum to one under the chosen convention and retain enough precision for the geometric means. Show the product and root for at least one criterion.
Changed-context transfer: If a criterion receives zero weight, weighted values become zero and the geometric-mean step needs a declared treatment. It is usually clearer to remove a truly irrelevant criterion and renormalise the remaining weights.
Limit and misconception check: The border is not an external benchmark. A positive distance means above the constructed border for that criterion, not objectively good performance.
Practical decision record: Calculate the accuracy border from 0.60 × 0.40 × 0.80 and take the cube root. Explain why the result lies between the three weighted values.
The signed distance is qᵢⱼ = vᵢⱼ − gⱼ. Positive q places the alternative in the upper approximation area for that criterion; negative q places it below the border. Distances are summed across criteria to give Sᵢ = Σqᵢⱼ.
Option A has approximate distances +0.023, −0.116 and +0.052, for S around −0.041. Option B has −0.177, +0.184 and +0.152, for S around +0.159. Option C has +0.223, −0.016 and −0.148, for S around +0.059. The descending ranking is therefore B, C, A.
The decomposition matters more than the final order. B wins through speed and cost despite lower accuracy. C leads accuracy but pays a large cost penalty. A remains near the border overall. A decision owner can now see the trade-offs that produced the ranking.
| Alternative | q accuracy | q speed | q cost | Total S | Rank |
|---|---|---|---|---|---|
| A | +0.023 | −0.116 | +0.052 | −0.041 | 3 |
| B | −0.177 | +0.184 | +0.152 | +0.159 | 1 |
| C | +0.223 | −0.016 | −0.148 | +0.059 | 2 |
Evidence check: Check that each q value equals weighted value minus the corresponding border and that the score equals the row sum. Keep a reproducible spreadsheet or script.
Changed-context transfer: If a stakeholder questions B, show the criterion contributions rather than only the rank. The discussion can then focus on whether speed and cost weights are justified.
Limit and misconception check: A negative total does not mean an alternative is unacceptable; it means it lies below the constructed borders in aggregate. Acceptance requires external constraints and judgement.
Practical decision record: Remove the cost contribution and compare the order. Then decide whether that scenario represents a legitimate change in priorities or an attempt to engineer a preferred result.
A positive signed distance is not a probability of success, percentage advantage or margin above a regulatory standard. It is the difference between one weighted normalised value and a geometric border created from the current alternatives.
Criterion signs can reveal a profile. An option may be above the border on two modest criteria and far below on one critical criterion. The total can still be positive, which is why decomposition and hard constraints must accompany the score.
Near-zero distances deserve attention to measurement uncertainty and rounding. If the underlying difference is smaller than data error or stakeholder disagreement, a precise rank may communicate more certainty than the evidence supports.
Evidence check: Present contribution plots or a signed-distance table with raw values, uncertainty notes and constraints. Report ties or practical equivalence when justified.
Changed-context transfer: If an option moves from −0.002 to +0.001 after a tiny data update, treat it as near the border rather than narrating a dramatic change from bad to good.
Limit and misconception check: MABAC supplies an ordinal ranking under inputs; it does not estimate outcome likelihood, causal effect or statistical significance.
Practical decision record: Rewrite three misleading statements—‘80 percent likely,’ ‘objectively above standard,’ and ‘proved best’—as accurate descriptions of the MABAC result.
Sensitivity analysis asks whether reasonable input changes alter the ranking. Vary one weight within a justified range while renormalising the others, test plausible measurement intervals, and examine scenarios rather than arbitrary perturbations.
MABAC’s min–max scaling and geometric border are both option-set dependent. Adding or removing an alternative can change normalised values and borders even when the old raw data remain fixed. A rank change should be investigated, not hidden.
The right question is not whether the ranking never changes. It is where it changes and whether those conditions are plausible. A robust decision can survive reasonable uncertainty; a fragile one needs more evidence, a narrower claim or a governance discussion.
Evidence check: Publish the tested ranges, scenarios and rank outcomes. Include at least one structural test: remove an option, add a plausible extreme or change a criterion definition.
Changed-context transfer: If B remains first until the accuracy weight exceeds 0.62, decision-makers can assess whether such a priority is credible. The threshold is more informative than one default rank.
Limit and misconception check: One-at-a-time sensitivity can miss interacting changes. Scenario or Monte Carlo approaches may be needed when several uncertain inputs move together.
Practical decision record: Create three scenarios: cost-priority, balanced and accuracy-priority. Record both ranking and contribution changes, then identify the smallest plausible change that reverses first and second.
The score is compensatory: strength on one criterion can offset weakness on another. That property is useful when trade-offs are genuinely permitted. It is dangerous when a minimum standard must be met.
A veto can be implemented as a feasibility screen before MABAC or as a separately reported non-compensable rule. For example, an option lacking required certification should be excluded or marked ineligible, not allowed to regain first place through low cost.
Thresholds should be defined before seeing the preferred option. Retrofitting a veto after ranking can be as manipulative as ignoring one. Governance requires an owner, rationale and appeal or review pathway.
Evidence check: List every hard constraint with source, threshold, test and responsible authority. Verify feasibility before normalisation and retain excluded options in an audit log.
Changed-context transfer: If a capacity floor becomes negotiable during an emergency, document the authorised change and rerun the decision. Do not silently convert the gate into a small penalty.
Limit and misconception check: Too many vetoes can leave no feasible alternatives and conceal the need to redesign the option set. A failed screen may indicate that the decision problem itself must change.
Practical decision record: Take the fictional options and add a minimum accuracy of 75. Apply the gate before MABAC, then compare the eligible set and explain why the result is not directly comparable to the three-option ranking.
Distinguish data, preference, frame and rule changes
One useful stability audit separates four kinds of change. Data change replaces an observed value with better evidence. Preference change revises a weight because the decision owner’s priorities changed. Frame change adds or removes an alternative or criterion. Rule change alters a hard constraint or the normalisation convention. These changes should never be described as if they were equivalent.
Suppose Option B’s speed falls from 9 to 8 after a corrected measurement. That is a data update and should be rerun with the same declared preferences. Suppose the team instead increases the accuracy weight after discussion. That is a preference update and needs a recorded rationale. If a fourth option enters, the ranges and borders change; that is a frame update. If the cost ceiling rises, eligibility changes before ranking; that is a rule update.
Keeping these versions separate helps a reviewer understand why a ranking moved. Without the distinction, a preferred outcome can be produced by quietly changing several inputs at once. A clean audit retains the earlier result, names the new evidence or authority, recalculates every dependent value and compares contribution changes rather than merely announcing a new winner.
Practical evidence check: version the raw matrix, criterion register, feasible-set screen, weight vector and calculation sheet together. Give the package a decision date and owner. A result copied into a slide without these inputs is a claim that cannot be independently tested.
Changed-context transfer: if the method is moved from supplier selection to public-policy prioritisation, the arithmetic can transfer but the governance cannot be assumed. Affected communities, distributional impacts and rights may require participation, disaggregated evidence and non-compensable safeguards that were not part of the supplier example.
Limit check: agreement between MABAC and another ranking method is not proof that the decision is correct. The methods may share the same weak matrix, weights or compensatory assumptions. Disagreement is useful when it exposes a reference-frame choice that decision-makers need to examine.
Decision record: state which result is being recommended, which reasonable scenario changes it, which constraints remain binding and who accepts the residual uncertainty. This closing record turns a calculation into an accountable recommendation without pretending that the formula carries authority on its own.
Weights can come from direct allocation, pairwise comparison, swing weighting or another elicitation method. Whatever the method, they should reflect the importance of moving across the actual criterion ranges, not abstract admiration for a criterion name.
Group decisions need a rule for combining views. Averaging weights can hide conflict between stakeholders who value different outcomes. Report the distribution of views, run stakeholder scenarios and identify criteria where disagreement drives the rank.
Avoid double counting. Cost, affordability and budget impact may overlap; accuracy, quality and reliability may measure related constructs. Correlated criteria are not automatically invalid, but their combined influence should be intentional.
Evidence check: Document who supplied weights, what information they saw, how disagreement was handled and whether ranges were understood. Test duplicated or correlated criteria.
Changed-context transfer: If the option range changes substantially, revisit weights elicited through swing weighting because the size of the swing has changed.
Limit and misconception check: A mathematically consistent weight vector can still represent an unfair process or exclude affected voices. Procedural legitimacy sits outside the formula but inside the decision.
Practical decision record: Ask two stakeholders to allocate weights independently and explain each allocation. Compare rankings under both rather than averaging immediately.
Add Option D with extraordinary speed but unacceptable safety certification. Its raw speed expands the maximum, changing old normalised speed values and the geometric border. If D is left in the matrix, it can reshape the ranking even though it is infeasible.
The correct first move is not to calculate faster. Apply the predeclared safety gate. If D fails, remove it from the eligible ranking and record the exclusion. Then rerun MABAC on the feasible set using current ranges.
Now suppose D gains certification but its speed measurement is uncertain. It becomes eligible, yet the range and border sensitivity should be tested. A scenario using lower and upper credible speed values reveals whether D’s inclusion creates a stable improvement or a fragile rank reversal.
Evidence check: Keep a before-and-after matrix showing the gate, rescaled values, borders and rank contributions. A reviewer should be able to distinguish eligibility change from score change.
Changed-context transfer: The same logic applies when a new technology, supplier or site enters later. Treat the option set as a versioned input and rerun all dependent steps.
Limit and misconception check: An extreme but feasible option can legitimately change the frame. Stability does not require old rankings to remain fixed when the decision set materially improves.
Practical decision record: Write two decision notes: one where D is ineligible and one where it is eligible but uncertain. State the evidence needed to move between them.
A reliable MABAC analysis has a visible sequence: frame the decision, screen feasibility, define criteria, collect comparable evidence, set directions, normalise, apply weights, construct borders, calculate signed distances, rank, decompose contributions and test sensitivity.
Every transformation should be reproducible. Preserve raw data, formulas, precision, option-set version and weight rationale. Presenting only the final score removes the very transparency that makes a multi-criteria method useful.
The wider Judgement route includes TOPSIS, VIKOR, COPRAS, WASPAS, MOORA, EDAS and MARCOS. Comparing methods can reveal which assumptions—distance, regret, proportionality, multiplication, average solution or reference alternatives—drive a decision.
Evidence check: An independent checker should be able to rebuild the ranking and obtain the same result. A stakeholder should be able to see which criteria and constraints explain it.
Changed-context transfer: For a new problem, transfer the audit sequence but rebuild the criteria, ranges, thresholds, evidence and weights. Do not carry a previous matrix into a different decision merely because the software is ready.
Limit and misconception check: MABAC does not replace legal authority, professional judgement, public participation or ethical review. It structures trade-offs under declared assumptions.
Practical decision record: Prepare a one-page decision sheet with purpose, feasible options, constraints, raw matrix, weights, border contributions, sensitivity thresholds, unresolved uncertainties and the accountable final decision-maker.
Questions readers ask
Is the border approximation area a real alternative?
No. It is constructed criterion by criterion as the geometric mean of weighted normalised values for the current alternatives.
Does a positive MABAC score mean an option is acceptable?
Not automatically. It means the alternative is above the constructed borders in aggregate. External requirements and veto conditions determine eligibility or acceptability.
Can adding an option change old rankings?
Yes. Min–max ranges and geometric borders depend on the option set. Additions and removals require a complete recalculation and stability check.
Should every requirement become a weighted criterion?
No. Non-negotiable legal, safety or capacity requirements are often better treated as gates so that good performance elsewhere cannot compensate for failure.
Practise, verify and continue
Independent practice: reproduce the fictional ranking using unrounded values. Then add an alternative with accuracy 95, speed 10 and cost 120. Recalculate the normalisation and border before applying any judgement about the new option. Record whether old ranks change and why.
Next, impose a maximum acceptable cost of 90 as a hard gate. Compare the feasible-set result with a calculation that leaves the over-budget option in the compensatory matrix. Explain which design matches the stated decision rule.
Related eduKate reading
TOPSIS ideal distances, normalisation and rank reversal
EDAS average-solution distances
MARCOS ideal and anti-ideal alternatives
VIKOR compromise solutions, regret and acceptable advantage
Authoritative sources and scope
Pamučar and Ćirović: original MABAC method paper
ScienceOpen record for the original MABAC paper
The sources establish the published MABAC method and its original logistics application. The matrix, weights and calculations in this guide are fictional and rounded for teaching. Method selection and final decisions remain context dependent.