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arxiv: 2604.09555 · v1 · submitted 2026-02-05 · 💻 cs.AI · math.OC

Linear Programming for Multi-Criteria Assessment with Cardinal and Ordinal Data: A Pessimistic Virtual Gap Analysis

Fuh-Hwa Franklin Liu , Su-Chuan Shih This is my paper

Pith reviewed 2026-05-16 07:39 UTC · model grok-4.3

classification 💻 cs.AI math.OC
keywords multi-criteria analysislinear programmingvirtual gap analysispessimistic assessmentcardinal dataordinal dataalternative rankingdecision support
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The pith

Two virtual gap analysis linear programs assess multi-criteria alternatives from a pessimistic viewpoint using both cardinal and ordinal data.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a two-step method for multi-criteria analysis that handles alternatives evaluated with both numerical measurements and ranked preferences. It introduces two linear programming models under the virtual gap analysis framework to generate pessimistic performance scores for each alternative. These scores support a prioritization step that eliminates the least favorable option. The approach seeks to limit the influence of subjective parameter choices that often affect traditional ranking techniques. A reader would care because the method claims to deliver consistent results across mixed data types without additional normalization steps.

Core claim

The paper claims that integrating two novel Virtual Gap Analysis linear programs into a two-step process produces consistent pessimistic scores for each alternative when both cardinal and ordinal data are present, allowing reliable prioritization and elimination of the weakest alternative.

What carries the argument

Virtual Gap Analysis (VGA) linear programs that compute pessimistic performance scores by measuring gaps between each alternative and the best possible values across criteria.

If this is right

  • Alternatives receive rankings derived directly from the pessimistic scores without separate weight estimation for each criterion.
  • The models scale to larger sets of alternatives because they rely on standard linear programming solvers.
  • Mixed data types can be processed in one framework, removing the need to convert ordinal ranks into numerical values beforehand.
  • Decision support systems can incorporate the method to produce repeatable elimination decisions across repeated assessments.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach might be tested on sequential decision problems where new alternatives arrive over time to see if the pessimistic scores remain stable.
  • One could compare the eliminated alternative against outcomes from established ranking methods on the same dataset to measure agreement rates.
  • Extending the models to include uncertainty bounds on the input data would show whether the pessimistic scores widen or stay narrow.

Load-bearing premise

The integration of cardinal and ordinal data into the two VGA linear programs produces consistent pessimistic scores without introducing new inconsistencies or requiring unstated normalization choices that affect the final elimination step.

What would settle it

Apply the two VGA models to a small test set of alternatives with explicit cardinal values and ordinal rankings, then check whether the alternative eliminated as weakest matches the one identified by direct comparison of the computed pessimistic scores.

Figures

Figures reproduced from arXiv: 2604.09555 by Fuh-Hwa Franklin Liu, Su-Chuan Shih.

Figure 1
Figure 1. Figure 1: The classifications of VGA-methods and VGA-models. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The w2c VGA-method with the owPT & ohPT VGA-models. [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Assessment solutions of DMU-A in Stage I [PITH_FULL_IMAGE:figures/full_fig_p027_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Assessment solutions of DMU-D in Stage II [PITH_FULL_IMAGE:figures/full_fig_p028_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Assessment solutions of DMU-1 in Stage I [PITH_FULL_IMAGE:figures/full_fig_p032_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Assessment solutions of DMU-1 in Stage II [PITH_FULL_IMAGE:figures/full_fig_p032_6.png] view at source ↗
read the original abstract

Multi-criteria Analysis (MCA) is used to rank alternatives based on various criteria. Key MCA methods, such as Multiple Criteria Decision Making (MCDM) methods, estimate parameters for criteria to compute the performance of each alternative. Nonetheless, subjective evaluations and biases frequently influence the reliability of results, while the diversity of data affects the precision of the parameters. The novel linear programming-based Virtual Gap Analysis (VGA) models tackle these issues. This paper outlines a two-step method that integrates two novel VGA models to assess each alternative from a pessimistic perspective, using both quantitative and qualitative criteria, and employing cardinal and ordinal data. Next, prioritize the alternatives to eliminate the least favorable one. The proposed method is dependable and scalable, enabling thorough assessments efficiently and effectively within decision support systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper presents a two-step method using two novel linear programming-based Virtual Gap Analysis (VGA) models to assess alternatives from a pessimistic perspective by integrating cardinal and ordinal data from quantitative and qualitative criteria. It then prioritizes the alternatives to eliminate the least favorable one, claiming the approach is dependable and scalable for use in decision support systems.

Significance. If the VGA models can be shown to produce consistent pessimistic scores that are invariant to reasonable normalizations of ordinal data, the method could offer a useful tool for multi-criteria decision making that reduces the impact of subjective biases in parameter estimation and handles mixed data types effectively.

major comments (2)
  1. [Abstract] Abstract: the central claim that the two VGA linear programs produce consistent pessimistic scores when jointly modeling cardinal and ordinal data is unsupported by any equations, constraints, or objective functions in the provided text. This prevents verification of whether ordinal data mapping (e.g., ranking-to-value conversion) introduces scaling choices that alter the feasible region or optimal pessimistic value.
  2. [Abstract] Abstract: no validation data, error analysis, comparison baselines, or scalability experiments are supplied to support the assertions of dependability and scalability, leaving the elimination step's reliability untested.
minor comments (1)
  1. [Abstract] Abstract: the phrasing 'dependable and scalable, enabling thorough assessments efficiently and effectively' repeats similar ideas and could be tightened for precision.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive comments. We address each major comment point by point below and indicate the revisions we will make to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the two VGA linear programs produce consistent pessimistic scores when jointly modeling cardinal and ordinal data is unsupported by any equations, constraints, or objective functions in the provided text. This prevents verification of whether ordinal data mapping (e.g., ranking-to-value conversion) introduces scaling choices that alter the feasible region or optimal pessimistic value.

    Authors: The abstract provides a high-level overview, while the full manuscript (Sections 3 and 4) presents the two VGA linear programming models in detail, including objective functions that minimize the virtual gap, constraints that jointly handle cardinal values and ordinal rankings via relative normalization, and the specific mapping that preserves consistency of pessimistic scores. The formulation uses gap-based constraints rather than absolute scaling to ensure invariance to reasonable ordinal conversions. We will revise the abstract to include a brief description of the LP structure and key consistency constraints. revision: yes

  2. Referee: [Abstract] Abstract: no validation data, error analysis, comparison baselines, or scalability experiments are supplied to support the assertions of dependability and scalability, leaving the elimination step's reliability untested.

    Authors: The manuscript prioritizes the methodological contribution of the two-step VGA procedure. We agree that empirical support is needed to substantiate the claims of dependability and scalability. In the revised version we will add a dedicated numerical experiments section containing validation examples, sensitivity/error analysis on the pessimistic scores, comparisons against established MCDM baselines, and scalability tests with varying numbers of alternatives and criteria. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the VGA LP derivation

full rationale

The paper introduces two novel linear programming models for Virtual Gap Analysis that integrate cardinal and ordinal data to produce pessimistic scores for alternatives, followed by a prioritization step to eliminate the least favorable one. No load-bearing step reduces by construction to its own inputs: the models are presented as new formulations whose feasible regions and objective values are determined by the input data and explicit constraints rather than by self-definition, fitted parameters renamed as predictions, or self-citation chains. The derivation chain remains self-contained against external benchmarks because the LP structure itself supplies the assessment mechanism without invoking unverified uniqueness theorems or ansatzes from prior author work.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The central claim rests on standard linear programming assumptions plus the unproven premise that the constructed virtual gaps correctly encode a pessimistic perspective across mixed data types. No explicit free parameters are named in the abstract, but LP models of this type typically introduce criterion weights or gap thresholds chosen to fit the data.

free parameters (1)
  • criterion weights or gap thresholds
    Typical in MCDM LP formulations; abstract does not specify but the method must assign relative importance or gap scaling to combine cardinal and ordinal inputs.
axioms (1)
  • domain assumption Linear programming can faithfully represent pessimistic virtual gaps for both quantitative and qualitative criteria
    Invoked when the paper states the VGA models tackle subjective bias and data diversity issues.
invented entities (1)
  • Virtual Gap Analysis (VGA) models no independent evidence
    purpose: To measure performance gaps from a pessimistic viewpoint and integrate cardinal/ordinal data
    Newly introduced constructs whose independent evidence would require external validation datasets not shown in the abstract.

pith-pipeline@v0.9.0 · 5437 in / 1374 out tokens · 30997 ms · 2026-05-16T07:39:20.166848+00:00 · methodology

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