Quality-Yield Index (QYI)

Updated 8 November 2025

Quality-Yield Index (QYI) is a composite metric that jointly measures volumetric output and quality to ensure balanced system performance.
It is operationalized using domain-specific formulations such as the harmonic mean or multiplicative combinations to address unique industry requirements.
QYI guides optimization in fields like agronomy, materials science, and combinatorial optimization by penalizing imbalanced contributions between yield and quality.

The Quality-Yield Index (QYI) is a composite metric designed to jointly quantify the productivity (yield) and output quality of systems, processes, or artifacts, where both the amount and the caliber of the outcome are critical. QYI appears in a variety of domains—from agronomy (e.g., sunflower oil production), fabrication of two-dimensional materials (e.g., graphene transfer), large-scale collaborative knowledge projects (e.g., Wikipedia evaluation), to the assessment of generative AI in combinatorial optimization—each requiring context-specific balancing of volumetric throughput and desired attribute performance. Below is an in-depth overview of the QYI concept, its mathematical foundations, implementations, and applications, accompanied by domain-specific instantiations as presented in recent literature.

1. Conceptual Foundations

QYI arises in contexts where univariate metrics (yield alone, or quality alone) are insufficient. High output of low-quality products, or excellent quality produced in negligible quantities, are both undesirable in most real-world applications. QYI addresses this by providing a measure that only rewards systems or processes that jointly optimize for both objectives, penalizing situations where one dimension falls short.

In principle, the QYI is defined as a function that combines quantitative yield (e.g., proportion of successfully completed products, solution pass rate, physical yield per area) and normalized quality (e.g., compositional purity, compliance with standards, expert-normalized utility). Several papers employ either direct multiplication, weighted combinations, or—for the most demanding applications—the harmonic mean to avoid hiding poor performance in either dimension (Chen et al., 9 Jun 2025).

2. Formalization and Mathematical Structure

General Format

The QYI is typically constructed as follows: $QYI = g(\text{Yield}, \text{Quality})$ where $g$ is a domain-specific function, often chosen to penalize imbalanced performance between quality and yield.

Harmonic Mean (HeuriGym, Combinatorial Optimization)

For agentic benchmarks such as HeuriGym, where large solution spaces and varying degrees of solution optimality are present, the QYI is defined as the harmonic mean of "yield" and "quality" (Chen et al., 9 Jun 2025): $QYI = \frac{2 \cdot \text{Quality} \cdot \text{Yield}}{\text{Quality} + \text{Yield}}$ with: $\text{Yield} = \frac{\hat{N}}{N}$ where $\hat{N}$ is the number of valid solutions, and $N$ is the total number of instances.

$\text{Quality} = \frac{1}{\hat{N}} \sum_{n=1}^{\hat{N}} \min\left(1, \frac{c_n^*}{c_n}\right)$

where $c_n^*$ is the cost (or value) of the expert baseline solution, and $c_n$ is the model’s cost. Values are capped at 1 for solutions exceeding expert quality.

Multiplicative Combination (Agronomy, Materials Science)

In plant production and 2D materials, QYI is conceptually implemented as a direct product: $QYI = Y \times f(Q)$ where $Y$ is the volumetric or area yield, and $f(Q)$ is a normalized, possibly weighted or transformed, quality function reflecting domain requirements (such as fatty acid composition, antioxidant content, or defect density) (Pereyra-Irujo et al., 2017, Burton et al., 2022, Cornelissen et al., 6 Oct 2025). The functional form of $f(Q)$ may involve direct normalization, utility mapping, or industry-specific preference curves.

Coverage-Quality Synthesis (Knowledge Bases)

For distributed content production (e.g., Wikipedia), QYI is operationalized by synthesizing an article’s prominence (abstracted as "yield" or coverage, e.g., via citation index or wikilinks) with a normalized quality score based on content features, correcting for language-specific standards (Lewoniewski et al., 22 May 2025):

Synthetic quality is normalized to top internal standards.
Citation index selects prominent articles for averaging.
QYI is thus represented as average synthetic quality of top-yield (most-cited) articles within a topic/language.

3. Domain-Specific Implementations

Combinatorial Optimization (HeuriGym Benchmark)

In code synthesis for combinatorial optimization, QYI addresses the dual requirement of producing correct (constraint-satisfying) code that is also of high optimization quality (e.g., cost minimization, solution optimality). The metric sharply penalizes models that succeed in only one dimension. For example, state-of-the-art LLMs on HeuriGym tasks achieved $QYI \approx 0.6$ , indicating substantial room relative to expert baselines (which score 1.0) (Chen et al., 9 Jun 2025).

Metric	Definition	Scale
Yield	Fraction of valid solutions	0–1
Quality	Mean relative solution quality	0–1
QYI	Harmonic mean of Yield and Quality	0–1

A QYI below 1 signifies either missed cases, or sub-expert performance on solved cases. This property allows QYI to be used in model selection, benchmarking, and progress tracking for agentic problem-solving systems.

Crop and Food Systems (Sunflower Model)

In agronomic modeling, QYI supports the simultaneous optimization of harvestable yield and component quality (e.g., oil percent, fatty acid profile, antioxidant capacity). While no explicit scalar QYI formula is set in (Pereyra-Irujo et al., 2017), the derived outputs (yield per area, targeted compositional metrics) form the basis for constructing customized indices: $QYI = GY \times f(Q), \quad f(Q) = \text{function of OA, LA, TC, T:L}$ Here, $GY$ is grain yield, and $f(Q)$ can be chosen to weight and normalize desired quality attributes for use-specific optimization (e.g., maximizing linoleic acid for processed foods, or tocopherol for oxidative stability).

Wikipedia and Collaborative Knowledge Bases

For evaluating coverage and quality across multilingual repositories, the QYI concept is realized via integration of a citation-based “yield” proxy (number of inbound links per article) and a normalized, feature-based synthetic article quality score (Lewoniewski et al., 22 May 2025). This allows comparison within and across language editions and topical categories.

2D Materials Synthesis

In advanced materials engineering, QYI is not expressed as a single scalar but as the co-localization of optimal process conditions in high-dimensional descriptor spaces (e.g., transfer yield and defectivity for graphene grown on various Cu facets) (Burton et al., 2022). The approach suggests constructing QYI as a function of area transfer yield and normalized quality metrics (e.g., Raman-derived defect metrics, electronic mobility), enabling holistic process optimization.

In-Field Analytics and Precision Viticulture

Spatially-resolved yield and quality mapping of crops (e.g., grapevines) enables the flexible aggregation of QYI at arbitrary units (e.g., per bunch, per zone) (Cornelissen et al., 6 Oct 2025). The index takes the form: $\text{QYI}_i = W_i \cdot f(Q_i)$ with $W_i$ as estimated weight (yield) and $Q_i$ as a grower-defined quality formula (e.g., $a\,\mathrm{Brix}_i - b\,\mathrm{Acidity}_i$ ). Such granular QYI mapping underpins zonal management, selective harvesting, and temporal tracking of agronomic interventions.

4. Properties, Constraints, and Interpretive Considerations

The utility of QYI is tightly linked to its mathematical coupling of quality and yield. Harmonic mean formulations prevent high scores when performance collapses in either dimension, reflecting the realistic necessity for balanced optimization. Multiplicative schemes (as in agronomy and materials science) can be linearly sensitive to deviations in either axis, necessitating careful normalization of quality scales relative to yield.

A key feature, especially in benchmarks such as HeuriGym, is that QYI is maximized only when both solution rate and per-instance performance align closely with expert standards. This sensitivity aids in revealing trade-offs, e.g., when interventions increase solution quality at the expense of feasibility rate or vice versa.

5. Applications and Implications

QYI has demonstrably broad application:

Machine-generated heuristic evaluation: Setting rigorous standards in open-ended coding and optimization tasks by accounting for both validity and optimality (Chen et al., 9 Jun 2025).
Crop system simulation and management: Guiding sowing or harvesting strategies optimized for both economic and nutritional objectives (Pereyra-Irujo et al., 2017).
Collaborative resource comparison: Enabling cross-lingual and cross-topical benchmarking of informational coverage versus quality (Lewoniewski et al., 22 May 2025).
Materials discovery and process engineering: Supporting descriptor-driven, multi-objective optimization in experimental science (Burton et al., 2022).
Precision agriculture: Facilitating the spatial prioritization of high-value zones for quality-targeted interventions and harvest logistics (Cornelissen et al., 6 Oct 2025).

A plausible implication is that as data granularity and process automation increase, QYI (or its domain-specific equivalents) will become a central metric in both research and operational contexts with competing objectives. The index also exposes systemic weaknesses: for example, in HeuriGym, even advanced LLMs do not approach expert-level joint performance, and many small Wikipedia editions have poor QYI outside a core topic set.

6. Comparison with Alternative Metrics

QYI fundamentally differs from single-axis success metrics (pass rates, average performance) and subjective scoring systems:

Metric	Multiple Solution Paths	Partial Quality Credit	Penalizes Imbalance	Domain Suitability
Accuracy / Pass@ $k$	✗	✗	✗	Closed-form, binary tasks
ELO/Subjective Score	✓	✓	✗	Subjective, open-ended tasks
QYI	✓	✓	✓	Optimization, multi-factored output

The selection of QYI is most appropriate where complex trade-offs exist, incremental improvements in either dimension matter, and both dimensions are essential to application value.

7. Limitations and Generalization

While QYI provides an interpretable and rigorous framework, the operational form must be contextualized:

Normalization and weighting of quality parameters may be application-specific; arbitrary composite functions risk masking domain-relevant features.
Construction of QYI in settings lacking expert benchmarks or with ambiguous quality definitions requires additional calibration and stakeholder consensus.
In multi-dimensional optimization, a scalar QYI may need to be supplemented with Pareto front analyses or multi-objective visualizations to avoid information loss.

This suggests that while QYI is not universally prescriptive, its structured approach to integrating yield and quality informs best practice in analytical benchmarking, resource optimization, and progressive automation across fields.

References:

(Pereyra-Irujo et al., 2017): Quantitative modeling of yield and oil quality in sunflower
(Burton et al., 2022): Process optimization of graphene synthesis and transfer
(Lewoniewski et al., 22 May 2025): Multilingual and multi-topic evaluation of Wikipedia via coverage-quality synthesis
(Chen et al., 9 Jun 2025): QYI as primary metric in agentic LLM benchmark for combinatorial optimization
(Cornelissen et al., 6 Oct 2025): Spatially-resolved grape yield and quality mapping for precision viticulture