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

This paper introduces a novel, scalable linear programming framework called Virtual Gap Analysis (VGA) that integrates cardinal and ordinal data to perform pessimistic, two-step multi-criteria assessments, thereby mitigating subjective biases and enabling efficient ranking and elimination of the least favorable alternatives in decision support systems.

The Gist

Imagine you are the manager of a fleet of delivery trucks. You want to know which truck is the best and which one is the worst so you can decide which one to keep and which one to retire.

But here's the catch: You can't just look at one thing. You have to look at:

• Fuel efficiency (a number, like miles per gallon).
• Driver safety record (a number, like accidents per year).
• Customer satisfaction (a rating from 1 to 5 stars).
• Brand reputation (a subjective score from a survey).

This is what experts call Multi-Criteria Assessment (MCA). It's like trying to grade a student not just on math scores, but also on their handwriting, their attitude, and their participation in class.

The problem with most existing methods is that they are like a biased teacher. They might say, "I think safety is twice as important as fuel," or they might struggle to compare a "5-star rating" with "gallons of gas." This leads to unfair rankings.

The Solution: The "Virtual Gap" Detective

This paper introduces a new, smarter way to rank these alternatives called Virtual Gap Analysis (VGA). Think of it as a high-tech, unbiased detective that uses math (specifically Linear Programming) to find the truth.

Here is how the new method works, broken down into simple steps:

1. The Two-Step "Worst-First" Strategy

Instead of trying to find the "Superstar" immediately, this method starts by finding the "Strugglers." It works in two stages:

• Stage 1: The "Worst Practice" Filter.
  Imagine a room full of people. The method asks: "Who is the least efficient?" It doesn't just look at the numbers; it looks at how much "virtual money" (a made-up currency used for comparison) you would have to spend to fix a bad truck.

  • If a truck is already perfect, the "cost to fix" is $0.
  • If a truck is terrible, the "cost to fix" is $100.
  • The method groups everyone who has a "fix cost" of $0 into a "Worst Group." These are the trucks that are already as bad as they can possibly be compared to their peers.

• Stage 2: The "Who's the Worst of the Worst?" Showdown.
  Now, we have a small group of the worst-performing trucks. We need to know which one is the absolute bottom.
  The method runs a second, more intense test on just this group. It asks: "If we take away the best truck from this group, who is left holding the bag?"
  The one with the highest "fix cost" in this final round is declared the Worst Alternative. You remove them from the fleet. Then, you repeat the process to find the next worst, and so on, until you have ranked everyone from best to worst.

2. The Magic of "Virtual Currency"

How does it compare apples (fuel) to oranges (customer ratings)?
The method invents a Virtual Currency.

• It assigns a "price" to every input and output.
• It calculates a "Virtual Gap": The difference between what the truck costs (inputs) and what it earns (outputs) in this virtual world.
• The Analogy: Imagine you are comparing a heavy truck and a light bike. The method says, "Okay, let's pretend 1 gallon of gas costs $10, and 1 star of satisfaction is worth $50." It balances the scales so you can compare them fairly without needing to convert everything into the same physical unit.

3. Handling "Subjective" Data (The Likert Scale)

What about the "Customer Satisfaction" rating (1 to 5 stars)?
The method treats these ratings like a ruler. It knows that a "5" is the top of the ruler and a "1" is the bottom. It doesn't guess what a "5" means; it simply knows it's the best possible position. It uses this to calculate how much "virtual money" is lost when a truck gets a "3" instead of a "5."

Why is this a Big Deal?

1. No Bias: Traditional methods often ask a human to decide, "Is safety more important than speed?" This new method lets the math decide the weights automatically based on the data. It's like a judge who has no personal feelings about the drivers.
2. Handles Mixed Data: It can easily mix hard numbers (dollars, tons) with soft numbers (ratings, surveys) without getting confused.
3. The "Pessimistic" View: Most methods try to be optimistic ("Look how good this truck is!"). This method is pessimistic ("Look how much this truck needs to improve"). This is actually more useful for decision-makers because it highlights the risks and the worst-case scenarios.
4. Scalable: It can handle a fleet of 10 trucks or 10,000 trucks without breaking a sweat.

The Bottom Line

This paper presents a new, robust tool for making tough decisions. Whether you are choosing the best university, the most efficient factory, or the safest airline, this method acts like a fair, mathematical referee.

It doesn't just tell you who won; it tells you exactly how far the losers are from winning and what specific changes they need to make to get there. It turns a messy, subjective debate into a clear, data-driven ranking.

Technical Summary

1. Problem Statement

Multi-Criteria Assessment (MCA) is essential for ranking alternatives (Decision Making Units or DMUs) based on multiple criteria. However, existing methods face significant limitations:

• Data Heterogeneity: Traditional methods struggle to handle decision matrices containing a mix of cardinal data (quantitative, continuous) and ordinal data (qualitative, Likert scales).
• Subjectivity and Bias: Methods like Multiple Criteria Decision Making (MCDM) and Stochastic Frontier Analysis (SFA) often rely on subjective weight assignments or statistical assumptions that introduce bias.
• Theoretical Flaws in DEA: Data Envelopment Analysis (DEA), a semi-parametric method, suffers from theoretical inconsistencies regarding unit measurement in objective functions and the use of "artificial goal weights" that do not guarantee efficiency scores within the feasible range $[0, 1)$.
• Perspective Limitations: Most methods focus on "best practice" (optimistic) assessments. There is a lack of robust, non-parametric frameworks for "worst practice" (pessimistic) assessments, which are crucial for identifying and eliminating the least favorable alternatives.

2. Methodology: The Pessimistic Virtual Gap Analysis (VGA)

The authors propose a novel, two-stage linear programming framework called Virtual Gap Analysis (VGA), specifically the $w2c$ method (Worst practice, Type 2 matrix, Perspective-c). This method integrates both cardinal and ordinal data without requiring subjective weighting or normalization.

Core Concepts

• Virtual Gap: The discrepancy between the total weighted virtual inputs and outputs of a DMU. It serves as a unit-less measure of inefficiency.
• Unified Goal Price ($\tau_o$): A virtual currency used to standardize the value of inputs and outputs, ensuring unit invariance.
• Two-Stage Process (Perspective-c):
  1. Stage I (Identification): Uses the Ordinal-Worst-Pure Technical ($owPT$) model to partition DMUs into "non-worst" and "worst" sets.
  2. Stage II (Ranking): Uses the Ordinal-Hypo-Pure Technical ($ohPT$) model to rank the "worst" DMUs identified in Stage I, identifying the single least favorable alternative.

Mathematical Formulation

The method utilizes a Primal-Dual Linear Programming approach for each DMU ($DMU_o$):

• Stage I ($owPT$ Model):

  • Objective: Minimize the Total Virtual Gap (TVG) or maximize the Total Adjustment Price (TAP).
  • Logic: It determines if a DMU can be improved by simultaneously increasing inputs and decreasing outputs (worst practice scenario).
  • Constraint: Handles ordinal data by constraining adjustments within Likert scale boundaries ($M^L, M^U$).
  • Outcome: DMUs with a virtual gap of $\Delta = 0$ are classified as the "worst" set ($E_{owPT}$). Those with $\Delta > 0$ are "non-worst" and ranked by the magnitude of their gap.

• Stage II ($ohPT$ Model):

  • Objective: Evaluate the "hypo-virtual gap" within the set of worst DMUs ($E_{owPT}$).
  • Logic: Excludes the DMU being evaluated from its own reference set to find the best among the worst.
  • Normalization: A two-step calculation derives a unified goal price ($\tau^*_o$) to normalize the virtual gap to the interval $[0, 1)$.
  • Outcome: The DMU with the largest positive hypo-virtual gap is identified as the absolute worst performer.

Key Mathematical Innovations

• Unit Invariance: The model ensures that changing measurement units (e.g., kg to tons) does not alter the ranking.
• Strong Complementary Slackness Conditions (SCSC): The authors rigorously verify that the primal and dual programs satisfy SCSC, ensuring the optimality and consistency of the solutions.
• Ordinal Handling: Unlike DEA, which struggles with ordinal data, VGA incorporates ordinal constraints directly into the linear programming formulation using penalty prices for boundary violations.

3. Key Contributions

1. Unified Framework for Mixed Data: The paper introduces a non-parametric method capable of processing decision matrices containing both cardinal (quantitative) and ordinal (qualitative/Likert) data simultaneously.
2. Pessimistic Assessment: It provides a rigorous mathematical foundation for "worst practice" analysis, allowing decision-makers to objectively identify and eliminate the least efficient alternatives.
3. Elimination of Subjectivity: By deriving "unified goal prices" through optimization rather than expert judgment, the method removes subjective bias and weighting issues common in MCDM and DEA.
4. Theoretical Correction: The authors demonstrate that their approach corrects fundamental theoretical errors in DEA formulations regarding unit consistency and the definition of efficiency frontiers.
5. Scalability: The method is designed for real-time processing and can handle large-scale decision matrices (e.g., 20+ rows, 200+ columns) efficiently using standard linear programming software.

4. Results and Validation

The authors validated the method using two case studies:

• Minimal Example (6 Laptops): A Type 2 decision matrix with mixed inputs (weight, brand perception) and outputs (sales, user satisfaction).
  • Result: The model successfully identified a ranking order ($A \succ G \succ H \succ B \succ K \succ D$) and pinpointed DMU-D as the worst performer with a significant hypo-virtual gap. The 2D visualization confirmed the virtual gap logic.
• Large-Scale Real Problem (29 Chinese Provinces): An energy efficiency assessment involving 3 inputs and 3 outputs (one ordinal).
  • Result: The method identified 11 "worst" provinces in Stage I. In Stage II, DMU-1 was identified as the least efficient alternative with the highest hypo-virtual gap ($0.426$). The model provided specific adjustment rates (e.g., reducing specific inputs by 46.2%) to reach the target frontier.

5. Significance and Implications

• Decision Support Systems (DSS): The method is highly suitable for integration into AI, IoT, and Big Data environments for real-time decision-making in complex, uncertain conditions (e.g., Industry 4.0, Smart Cities).
• Robustness: The approach is immune to data heterogeneity and does not require the assumption of uniform DMU structures, a major limitation of traditional DEA.
• Actionable Insights: Beyond ranking, the model provides specific adjustment rates (how much to increase/decrease inputs/outputs) and identifies benchmark peers for underperforming units.
• Future Applications: The framework is extensible to handle incomplete, ambiguous, or imprecise data, offering a versatile tool for modern multi-criteria decision-making challenges.

In conclusion, this paper presents a mathematically rigorous, non-parametric alternative to existing MCA methods, specifically solving the problem of ranking alternatives in the presence of mixed data types while eliminating subjective bias through a pessimistic virtual gap analysis.