New tool PySensMCDA enables sensitivity analysis in multi-criteria decision problems

A new version of an open-source Python library is reshaping how researchers and decision-makers can trust the rankings produced by multi-criteria decision analysis, addressing one of the field’s most persistent weaknesses: the arbitrary choice of criterion weights. The software, known as PySensMCDA, has been updated to version 2.0 in the journal SoftwareX by Bartosz Paradowski and Wojciech Sałabun, and the release represents a substantial conceptual expansion of what sensitivity analysis in decision-making problems can achieve. Rather than merely asking how a ranking changes when weights are nudged, the new version flips the question around, asking instead what weights would have to exist for a given alternative to win, and how much of the entire plausible weight space supports each candidate.

Multi-criteria decision analysis, or MCDA, is the mathematical backbone of countless real-world decisions, from selecting suppliers and infrastructure projects to ranking medical treatments and environmental policies. Methods such as TOPSIS or PROMETHEE combine performance scores across several criteria—cost, quality, delivery time, reliability—into a single ranking. But every one of these methods requires weights that express how important each criterion is relative to the others. In practice, these weights are frequently unavailable, disputed among stakeholders, or expected to drift over time. When a ranking rests on weights nobody truly measured, its apparent precision can be misleading. Version 1.0 of PySensMCDA tackled this with classical perturbation analysis: adjust an element of the decision problem and observe how the ranking responds. Version 2.0 adds the inverse direction, along with a suite of tools designed to summarise many rankings at once, regardless of how they were produced.

The update is organised around four new submodules. The first handles weight space sampling, separating the assumptions made about criterion weights from the analyses that consume them. It offers uniform and Dirichlet sampling of the weight simplex, analytic sampling under an ordinal ranking of criteria, exhaustive simplex grids, and the Hit-And-Run algorithm for sampling under arbitrary linear constraints—a technique originally developed for efficient Monte Carlo generation of points over bounded regions. Decision-makers often hold only partial preference information: an ordering of criteria importance, bounds on individual weights, bounds on trade-off ratios, or general linear constraints. Version 2.0 expresses all four types through a single constraint builder, producing plain numerical arrays that can feed any downstream analysis in the package.

The second submodule implements the stochastic multicriteria acceptability analysis family, better known as SMAA, a methodology developed for group decision-making under uncertain or missing weight information. SMAA works by drawing millions of random weight vectors from the admissible space and computing, for each alternative, the share of weight space in which it attains each possible rank. These are the rank acceptability indices, reported with Monte Carlo standard errors. The implementation goes considerably further, offering central weight vectors that describe the “typical” preference profile supporting each alternative, favourable weight ranges, confidence and cross-confidence factors, holistic and k-best-ranks measures, and support for ordinal criteria through SMAA-O.

Perhaps the most technically striking addition is an exact route for deterministic problems. For an additive value function with fixed criteria values, the acceptability index is not estimated by simulation but computed exactly, as a ratio of polytope volumes—the volumes of geometric regions defined by linear inequalities. This serves two purposes. It provides a reference free of sampling error against which the required number of simulation iterations can be judged: in validation tests, the Monte Carlo and exact routes agree to 0.003 percentage points at 200,000 iterations. More importantly, it resolves a case that random sampling can never reach. An alternative whose favourable weight set is lower-dimensional—confined to a slice of zero volume within the weight space—has an acceptability of exactly zero, yet possesses a well-defined central weight vector. Simulation will never visit such a region, but the exact computation identifies it, distinguishing an alternative that can never win from one that wins only under a degenerate weight configuration. Two components of this exact route remain computationally inexpensive regardless of problem size: a linear-programming feasibility test that eliminates dominated alternatives, and the true favourable weight ranges obtained from just twice as many linear programs as there are criteria.

The third submodule implements the Exhaustive Objective Ranking Solution, or EORS, a recently proposed method that explores the admissible weight space systematically rather than by sampling. For each alternative it reports the preference value attained most often across the explored space, together with degrees of confidence describing how concentrated that outcome is. The release also incorporates a newly published local sensitivity extension, based on simplex-projected finite differences around the weight configuration supporting each global result, aggregated into a criterion-level sensitivity index and an alternative-level fragility index. The fourth submodule collects robustness metrics that summarise a distribution of ranking outcomes: ranking stability and balance point coefficients, pairwise winning indices, and decision entropy for both choice and ranking problems.

The architectural decision that ties the update together is a single, method-agnostic calling convention: any decision method can be supplied as a simple function taking a decision matrix, a weight vector, and a list of criterion types, and returning preference values. Any method from the companion pymcdm library satisfies this contract directly, as does any plain callable, with no inheritance or adapter code required. The consequence is significant. Because the ranking machinery underlying SMAA compares preference values and nothing else, substituting an arbitrary method yields SMAA-TOPSIS, SMAA-PROMETHEE, or SMAA applied to any user-defined method without additional code. Reference implementations of SMAA in R and Java are restricted to the additive value model, so this generalisation extends the reach of the entire SMAA family rather than merely reimplementing it. One caveat is documented explicitly: for methods with internal normalisation, the acceptability indices retain their probabilistic interpretation, but the central weight vector becomes a descriptive centre of gravity rather than an estimate of utility weights.

The power of the approach is illustrated with a supplier selection problem involving five alternatives and four criteria: unit cost and lead time, both of cost type, and quality audit score and delivery reliability, both of profit type. Evaluated with TOPSIS under equal weights, the problem yields the ranking A3 first, followed by A1, A5, A2 and A4. But suppose decision-makers cannot agree on weights, and instead agree only that quality is the most important criterion, followed by cost, reliability and lead time, with quality carrying between 25 and 50 percent of the total weight, and criteria values known only within 5 percent. Under these more realistic assumptions, the conclusion changes dramatically. Alternative A2, only fourth under equal weights, emerges as the most preferred option for 64 percent of the admissible weight vectors, with a confidence factor of 89 percent indicating the measurements are accurate enough to sustain that conclusion. Alternative A4 is never preferred at all. The recommended alternative under equal weights is not the one supported by the majority of plausible preferences.

The robustness metrics sharpen this picture further. When the entire weight simplex is explored without any preference constraints, every alternative eventually visits every position, and the ranking stability coefficient is zero throughout—under complete ignorance, no position is stable. Restricting the analysis to the preferences the decision-makers consider plausible separates the alternatives meaningfully. Decision entropy quantifies the same gain from a different angle: normalised choice entropy falls from 0.585 to 0.393, and ranking entropy from 0.625 to 0.315, once the stated preference information is imposed. This provides a direct, quantitative answer to a question decision analysts are often asked—whether further effort to elicit preferences is actually worth it.

Validation of the new implementation is unusually rigorous for scientific software. Both founding SMAA papers report complete input data and results, allowing them to serve as regression fixtures: the implementation reproduces their acceptability indices, confidence factors, central weight vectors, full rank acceptability matrices and holistic indices to within one percentage point, and this agreement is enforced as an automated test rather than reported once. The constrained Hit-And-Run sampler reproduces the distribution of the analytic ordered-simplex sampler with which it shares no code. The test suite comprises 371 tests covering 98 percent of the new modules. The software is released under the MIT licence on GitHub, with full documentation hosted online, requiring Python 3.10 or newer along with standard scientific libraries including numpy, scipy and pandas.

The broader impact is a shift in what a practitioner can honestly state about a recommendation. Instead of presenting a ranking derived from weights that were never measured, analysts can now report how much of the admissible weight space supports each alternative, which preference profiles would justify it, and whether the underlying measurements are precise enough to distinguish candidates at all. For methodological researchers, the update makes two recently introduced method families—EORS and the stability coefficients—available in a maintained implementation alongside established methods, on a common interface permitting direct comparison. Future releases may extend the framework toward outranking-based SMAA variants using ELECTRE-type methods, and may bring fuzzy variants of iterative compromise ranking analysis into the same interface. For a field in which weight elicitation has long been the weakest link between mathematical rigour and practical credibility, version 2.0 of PySensMCDA offers something close to a completeness check: a way of knowing not just which alternative wins, but how much of the decision space actually agrees.