Solving non-quadratic Matrices in Assignment Problems with an improved Version of Vogel's Approximation Method
Bibliographic record
Abstract
The efficient allocation of tasks to vehicles in a fleet of self-driving vehicles (SDV) becomes challenging for largescale systems (e.g. more than hundred vehicles). Operations research provides different methods that can be applied to solve such assignment problems. Integer Linear Programming (ILP), the Hungarian Method (HM) or Vogel’s Approximation Method (VAM) are frequently used in related literature (Paul 2018; Dinagar and Keerthivasan 2018; Nahar et al. 2018; Ahmed et al. 2016; Korukoglu and Balli 2011; Balakrishnan 1990). The underlying paper proposes an adapted version of VAM which reaches better solutions for non-quadratic matrices, namely Vogel’s Approximation Method for non-quadratic Matrices (VAM-nq). Subsequently, VAM-nq is compared with ILP, HM and VAM by solving matrices of different sizes in computational experiments in order to determine the proximity to the optimal solution and the computation time. The experimental results demonstrated that both VAM and VAM-nq are five to ten times faster in computing results than HM and ILP across all tested matrix sizes. However, we proved that VAM is not able to generate optimal solutions in large quadratic matrices constantly (starting at approx. 15×15) or small non-quadratic matrices (starting at approx. 5×6). In fact, we show that VAM produces insufficient results especially for non-quadratic matrices. The result deviate further from the optimum if the matrix size increases. Our proposed VAM-nq is able to provide similar results as the original VAM for quadratic matrices, but delivers much better results in non-quadratic instances often reaching an optimum solution. This is especially important for practical use cases since quadratic matrices are rather rare.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.003 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".