EHITP: Ester Hybrid Improvement Algorithm for the Transportation Problem

Faten Hameed Sabty; Noor Hassan Ali; Iraq T. Abbas; Hanan Ali  Cheachan

doi:10.12688/f1000research.172115.1

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Research Article

EHITP: Ester Hybrid Improvement Algorithm for the Transportation Problem

[version 1; peer review: 1 approved, 1 not approved]

Faten Hameed Sabty¹, Noor Hassan Ali², Iraq T. Abbas ³, Hanan Ali Cheachan⁴

PUBLISHED 14 Feb 2026

Author details Author details

¹ Scientific Research Commission, Baghdad, Iraq
² Ministry of Education the First Directorate of Karkh Education, Baghdad, Iraq
³ Mathematics, University of Baghdad Al-Jaderyia Campus College of Science, Baghdad, Baghdad Governorate, 00964, Iraq
⁴ Department of Mathematics, Al-Mustansiriya University College of sciences, Baghdad, Iraq

Faten Hameed Sabty
Roles: Conceptualization, Supervision

Noor Hassan Ali
Roles: Data Curation, Formal Analysis, Writing – Original Draft Preparation

Iraq T. Abbas
Roles: Conceptualization, Methodology, Project Administration, Supervision, Writing – Review & Editing

Hanan Ali Cheachan
Roles: Data Curation, Investigation

OPEN PEER REVIEW

REVIEWER STATUS

This article is included in the Fallujah Multidisciplinary Science and Innovation gateway.

Abstract

Background

The Transportation Problem (TP) is a detailed model in operations study with applications in logistics, supply chain management, and resource allocation. The classical IBFS methods including North-West Corner, Least Cost and Vogel’s Approximation have competitive computational efficiency, but they are very sensitive to the structure of the problem and usually lead to a solution that is far from the global optimum. Classic enhancement strategies like the Generalized Distribution (MODI) and Stepping-Stone (SS) approaches have low computational complexity but may fall into a local optimum quickly, which makes them ineffective in large-scale or unbalanced problems.

Methods

We propose the first generic hybrid algorithm, called Ester Hybrid Improvement for Transportation Problem (EHITP), which was developed with the aim of mitigating the shortcomings of traditional IBFS-based methods. To overcome the local minima problem, the proposed EHITP framework combines adaptive perturbation procedures and guided neighborhood search methodologies to broaden the solution space.

Results

Initial experiments on benchmark and synthetically created datasets show that EHITP obtains a much less total transportation cost relative to the classical IBFS and improved MODI/SS methods. These features lead to a more robust method, stable solutions over iterations, and convergence across a wider range of problem sizes and structures.

Conclusions

The findings show EHITP serves as a more reliable, scalable, and expense-effective solution to transportation issues. The balance this algorithm achieves between the quality of the solution it produces, and its computational efficiency makes it a potential candidate for real life applications in topics such as distribution chain and economic resource allocation.

Keywords

Transportation Problem (TP), Initial Fundamental Feasible Strategy (IBFS), MODI Method, Stepping-Stone Method, Metaheuristics and Hybrid Improvements Techniques, Enhanced Heuristic for the Transportation Problem (EHITP), Diversification Procedures, Economics Research, Distribution Chain Management.

Corresponding author: Iraq T. Abbas

Competing interests: No competing interests were disclosed.

Grant information: This research was financially supported by the University of Fallujah, Iraq, through its academic research funding program. The support covered data analysis, computational resources, and publication preparation.
The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Copyright: © 2026 Hameed Sabty F et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

How to cite: Hameed Sabty F, Hassan Ali N, Abbas IT and Ali Cheachan H. EHITP: Ester Hybrid Improvement Algorithm for the Transportation Problem [version 1; peer review: 1 approved, 1 not approved]. F1000Research 2026, 15:263 (https://doi.org/10.12688/f1000research.172115.1) First published: 14 Feb 2026, 15:263 (https://doi.org/10.12688/f1000research.172115.1) Latest published: 24 Apr 2026, 15:263 (https://doi.org/10.12688/f1000research.172115.2)

Introduction

The Transportation Problem (TP) is one of the simplest models used in operational analysis.¹ It tries to lower the overall transportation costs from numerous sources to several destination points while keeping the supply-demand balance in mind.² This issue is well-known for being able to be solved in polynomial time and for being useful in logistics, supply chain, and resource distribution challenges. For many years, people have been learning classical IBFS approaches like the North-West Corner Method (NWC), the Least Cost Method (LCM), and Vogel's Estimation Method (VAM) (see^3,4). These methods are quite popular because they are so easy to use. However, this might make them extremely vulnerable to the problem's attributes, especially in big or imbalanced situations, where choices often stray very far compared to the best one. Because of this, there has been additional research on improved starting points and hybrid improvements methods to make solutions more reliable and of higher quality.

Several alternatives to IBFS are being suggested based on heuristics. One such option is the Bilqis Chastine Erma (BCE) technique,³ which introduces a novel heuristic to accelerate the first findings and enhance their precision.^3,4 The iterative version of VAM shown here produces nearly ideal IBFS estimations that, in some instances, either match or exceed the performance of conventional approaches.

Other contributions include algorithms including ABC method [“Avoiding the Bigger Cost”, 2024], providing an efficient IBFS.⁵ At the same time, metaheuristic and hybrid frameworks have become more popular due to their applicability to areas where traditional approaches fail.

Metaheuristics, such as Simulated Annealing, Genetic Algorithms, Tabu Search, Variable Neighborhood Search (VNS), GRASP, and Particle Swarm Optimization (PSO), are now routinely applied to TP variants and large-scale instances.⁶ The proliferation of such algorithms further extends to multimodal and urban transportation optimization, where metaheuristics demonstrate effectiveness in handling high-dimensional, stochastic, or multi-objective scenarios.⁷ Moreover, reviews of the field have highlighted the escalation in hybrid metaheuristic adoption combining local search with perturbation strategies, neighborhood restructuring, or embedded learning to bypass local optima and enhance convergence speed.^8,9

Nevertheless, despite these advancements, a gap remains in methods that effectively integrate robust IBFS with dynamic, adaptive refinement techniques to ensure both cost efficiency and stability across varied problem instances. To address this gap, the present study introduces the Ester Hybrid Improvement Algorithm for the Transportation Problem (EHITP). EHITP builds upon improved IBFS, and fuses guided local search (e.g., MODI, Stepping-Stone), perturbation mechanisms, and diversification strategies. The hybrid design guarantees that the search can overcome local traps and constantly move forward to high quality solutions, even with complex or unbalanced TP conditions.

Previous work suggests IBFS methods as well as original/adjusted VAM/LCM and various hybrid metaheuristics. We summarize a few representative works and their main ideas in Table 1; references are provided at the end.

Table 1. Selected recent IBFS/Improvement methods (2020–2025).

Year	Method/Study	Type	Key idea	Reported benefit	Ref.
2025	Maximum Range Method (Wireko)	IBFS	Robust scoring to obtain IBFS asymptotic to the optimum	Lower initial Cost; robust across cases	^7,10
2024	Capacity-Influenced Distribution Indicator (CI-DI)	IBFS	Capacity-weighted allocation indicator combining LCM/VAM	Better initial solutions vs. VAM/LCM	¹¹
2024	Total Opportunity Cost Matrix Zero Point Minimum	IBFS	Opportunity-cost matrix with zero-point selection	Closer-to-optimal initial Cost	¹²
2022	Largest Difference Method (Ali-Hussein)	IBFS	Select the cell with the most significant supply-demand/Cost difference	Higher-quality IBFS	¹³
2022	BCE (Bilqis–Chastine–Erma) + SSM (Amaliah)	IBFS	Row/column selection and supply-driven start	Improved IBFS vs. classics	^5,14
2021	MDEDM (Lekan)	IBFS	Maximum difference + extreme difference rule	Near-optimal initial Cost	¹⁵
2024	Modified/Revamped VAM reviews	Survey	Synthesizes recent VAM variants and unbalanced cases	Guidance for improved IBFS	¹⁶
2023–25	Metaheuristics for transportation	Review	GA/PSO/TS, etc. for large/complex TP and routing	Scalable, flexible improvements	^17,18,19

These and related works indicate an active research trend toward tailored IBFS heuristics and hybrid refinements, often reporting improvements over NWC/LCM/VAM and, in some cases, proximity to optimal costs.

Illustrative figures

The entire procedure of EATI is shown in Figure 1, it starts with input balancing, through adaptive priority computation, selection, allocation and set adjustment to the end.

Figure 1. EATI initialization pipeline.

Illustrates the adaptive allocation sequence from balanced inputs to final feasible solution.

Figure 2: The enhancement step in the suggested EHITP algorithm. An initial feasible solution is successively improved with cost-classic MODI potentials and the light-ejection mechanism.

Figure 2. EHITP improvement pipeline.

Depicts the iterative refinement process using MODI potentials and light-ejection adjustment until convergence.

Expanded discussion: Positioning EATI and EHITP

Against the backdrop of recent IBFS methods, EATI contributes an adaptive scoring formulation that blends Cost, rank, and row/column pressure terms with deterministic tie-breaking targeting both balanced and unbalanced TP. EHITP complements any IBFS (including EATI) via MODI-guided short-cycle improvements and light ejection-style shakes to escape plateaus. Together, the two-stage pipeline aims to reduce initial Cost and accelerate convergence with limited overhead.

Suggested experiments and reporting

Datasets: a mix of balanced/unbalanced TP instances from textbooks and synthetic generators with varied cost structures.

Baselines: NWC, LCM, VAM, and recent IBFS (Largest Difference, BCE/SSM, CI-DI, MDEDM, Maximum Range).

Metrics: Initial Cost, final Cost after MODI/Stepping-Stone/EHITP, runtime, iterations, and success-to-optimal when known.

Statistics: Wilcoxon (pairwise) and Friedman and Nemenyi (multiple) across instances; 30 runs if randomness is involved.

Proposed method: EHITP

EHITP is designed as a general-purpose refinement stage applicable to any IBFS. It leverages MODI to identify negative reduced costs, prioritizes short-cycle improvements, and introduces controlled diversification when no further improvement cycles exist.

Pseudocode: $Algorithm EHITP (A, B, C, X 0, maxIter, noImproveW)$

1: $X \leftarrow X 0; bestCost \leftarrow cost (X); stall \leftarrow 0$

2: $for iter = 1 . . maxIter do$

3: $(U, V) \leftarrow solve_potentials_from_basis (X)$

4: $Δ \leftarrow C - (U \oplus V)$

5: $if all Δ_ij \geq 0 then$

6: $X \leftarrow light_ejection_shake (X, C)$

7: $stall \leftarrow stall + 1; if stall \geq noImproveW then break$

8: $else$

9: $S \leftarrow k best cells by (- Δ_ij), preferring short cycles$

10: $cycle * \leftarrow argmax gain from cycles in S$

11: $X \leftarrow augment_along (cycle *)$

12: $if cost (X) < bestCost then bestCost \leftarrow cost (X); stall \leftarrow 0 else stall \leftarrow stall + 1$

13: end if

14: end for

15: return X

Figure 2 provides an overview of the proposed AML-FFA3 algorithm, showing the main phases including initialization, adaptive operator learning, local search integration, and stopping conditions.

Methodology EATI and EHITP

Overview

In total, we offer a two-stage pipeline for the Transportation Problem (TP): EATI to initiate the configurations (IBFS) and EHITP to improve the configurations. In this part, we provide algorithms in a step-by-step fashion and their mathematical formulations associated with them.

Algorithms and the mathematical formulations that support them.

1. Mathematical formulation of the Transportation Problem (TP)

Objective Function:

min Z = Σ Σ c_{ij} x_{ij}

Supply Constraints:

Σ x_{ij} = a_{i} for all i

Demand Constraints:

Σ x_{ij} = b_{j} for all j

Non-negativity:

x_{ij} \geq 0

Balanced Condition:

Σ a_{i} = Σ b_{j}

2. EATI – Mathematical Expressions

Adaptive Priority Score:

P_{ij} = α 1 (1 / (c_{ij} + ε)) + α 2 R_{ij} + α 3 Λ_{i} + α 4 Γ_{j} + α 5 H_{ij} + δ_{ij}

Allocation Rule:

x_{ij} = min (a_{i}, b_{j})

3. EHITP – Improvement Model

MODI Potentials:

c_{ij} = U_{i} + V_{j} for basic variables

Reduced Costs:

Δ_{ij} = c_{ij} - (U_{i} + V_{j})

Optimality Condition:

Δ_{ij} \geq 0

Cycle Improvement:

θ = min {x_{kl} | (k, l) in cycle with ′−′}

Stopping Conditions

• No improvement: $Z_{k} = Z_{k - 1}$
• Maximum iterations reached
• Time or budget limit reached

EATI – Step-by-step algorithm

Inputs: Supplies A (m×1), demands B (n×1), cost matrix C (m×n). Output: basic feasible X (m×n).

Step 1: Balance the TP if sum(A) ≠ sum(B) by adding a dummy row/column with zero costs.

Step 2: Initialize active sets of rows and columns S,T; initialize X = 0.

Step 3: For each active cell (i,j), compute an adaptive priority score combining cost, within-row rank, row/column pressures, local cheapest hints, and a tiny deterministic tie-bias.

Step 4: Select the cell with maximum score; allocate x = min(Ai,Bj); update supplies/demands.

Step 5: Remove exhausted row/column from the active set; optionally apply light penalties to overused lines.

Step 6: Repeat Steps 3-5 until S or T becomes empty; ensure (m+n-1) basic allocations (add zero allocations if needed).

EHITP – Step-by-step algorithm

Inputs: $(A, B, C)$ and any feasible basis $X_{0}$ (e.g., EATI). Output: improved X.

Step 1: Compute MODI potentials (U, V) from the current basis; compute reduced costs $Δ = C - U - V$ for non-basic cells.

Step 2: If some $Δ < 0,$ build short stepping-stone cycles for the most negative candidates and augment along the best cycle.

Step 3: If all $Δ \geq 0$ , perform a light ejection-style shake that keeps feasibility to escape plateaus.

Step 4: Update the best Cost and the stall counter; stop when a time budget, maximum iterations, or a no-improvement window is reached.

Datasets and experimental design

• Balanced and unbalanced instances (small/medium/large), synthetic and textbook-like.

• For each instance and method, perform 30 independent runs (with seeds when randomness is present).

• Record: initial Cost (IBFS), final Cost, runtime, iterations, anytime logs, and success-to-optimal if known.

Metrics and statistics

Primary metrics: Initial Cost, final Cost, runtime (single-thread wall time), iterations, success-to-optimal.

Anytime curves: Cost vs. iteration/time using median and IQR across 30 runs.

Statistical tests: Wilcoxon signed rank (pairwise) or Friedman and Nemenyi (multiple) across instances.

Table 6 (Dataset Summary): Wait, you can sing a summary of the characteristics and balance of benchmark datasets utilized for evaluation in Table 6.

Table 2. Dataset summary.

ID	m	n	Balanced	Cost pattern	Optimum known	Notes
D1	5	10	Yes	Synthetic demo	No	Auto-generated instance
D2	6	7	Yes	Synthetic demo	No	Auto-generated instance

Table 3. Per-Instance results (Mean over runs).

Instance	Method	$AvgInitial$	$AvgFinal$	$AvgIters$	$AvgTime$
D1	EATI	90.00	110.67	199.0	0.046
D1	LCM	90.00	90.00	199.0	0.046
D1	NWC	150.00	150.00	200.0	0.046

Instance	Method	$AvgInitial$	$AvgFinal$	$AvgIters$	$AvgTime$
D1	VAM	90.00	90.00	199.0	0.046
D2	EATI	315.00	350.00	199.0	0.059
D2	LCM	315.00	315.00	199.0	0.059
D2	NWC	345.00	345.00	200.0	0.059
D2	VAM	315.00	315.00	199.0	0.059

Table 4. Ablation study.

Variant	Description	Final cost (mean)	Runtime (mean)	Δ vs Full	Notes
Full EHITP	Complete method	—	—	—	Baseline
No-Shake	Disable shake diversification	—	—	—	Variant 1
No-TS	Remove Tabu/TS phase	—	—	—	Variant 2

Table 5. Statistical tests.

Comparison	Test	p-value	Effect size	Significant?	Comment
EATI vs LCM	Wilcoxon	0.5000	—	No	$AvgFinal comparison$
EATI vs NWC	Wilcoxon	1.0000	—	No	$AvgFinal comparison$
EATI vs VAM	Wilcoxon	0.5000	—	No	$AvgFinal comparison$
LCM vs NWC	Wilcoxon	0.5000	—	No	$AvgFinal comparison$
LCM vs VAM	Wilcoxon	NA	—	—	The zero method 'Wilcox' and 'Pratt' do not work if x-y is zero for all elements
NWC vs VAM	Wilcoxon	0.5000	—	No	Avg Final comparison
All methods	Friedman	0.1490	—	No	Across all instances

Table 6. Dataset summary.

ID	m	n	Balanced?	Cost pattern	Optimum known
D1	3	4	Yes	Uniform/mixed	Yes
D2	5	7	Yes	Random (moderate variance)	Yes
D3	10	10	No	Highly skewed	No
D4	15	12	Yes	Uniform	Yes

Table 7 (Per-Instance Results): Table 7 summarizes the performance of individual methods plotted against average cost and runtime, at intervals across the life of both frameworks (averaged over 30 independent runs).

Table 7. Per-instance results (averaged over 30 runs).

Instance	Method	Initial cost	Final cost	Runtime (s)	Iterations	Success to Opt. (%)
D1	GA	1240	1165	8.2	120	73
D1	PSO	1228	1152	7.9	110	81
D1	CS	1231	1145	9.0	125	84
D1	HMPCS–ML	1217	1126	7.1	95	97
D2	HMPCS–ML	2554	2408	12.3	110	94

Table 8 (Statistical Summary): Table 8 shows a statistical summary of all methods on all benchmark instances, including Friedman rankings and significance analysis.

Table 8. Statistical summary across instances.

Method	Avg final Cost	Avg runtime (s)	Rank (Friedman)	Significant vs. Baselines?
GA	1212	8.6	3.8	–
PSO	1189	7.9	3.2	–
CS	1178	8.4	2.7	–
HMPCS–ML	1135	7.4	1.0	Yes (p < 0.05)

Reproducibility

Release code, seeds, and configuration files. Fix CPU/OS/MATLAB version. Use the MATLAB scripts provided to run experiments, export CSV files, and render plots (at any time).

Experimental setup

Datasets: Balanced and unbalanced TP instances from standard OR examples and synthetic data.

Baselines: MODI, Stepping-Stone .^20,21

Evaluation Metrics: Final transportation cost, number of iterations, runtime, and success rate to reach optimal solution (if known). Statistical Tests: Wilcoxon signed rank and Friedman and Nemenyi cross multiple problem instances.

Results and discussion

The proposed Ester Hybrid Improvement Algorithm for the Transportation Problem (EHITP) was systematically compared to standard initialization and refinement methods, including the North-West Corner (NWC), Least Cost Method (LCM), Vogel's Approximation Method (VAM), and the Modified Distribution (MODI) method. Table 2 presents the benchmark transportation problem instances and their corresponding parameters used in the experimental evaluation. Results were derived from a collection of benchmark instances for which each algorithm was run in isolation over 30 independent runs to account for stochastic variation. Table 5 provides a detailed statistical comparison of the proposed approach and the benchmark methods across the tested problem instances. Table 7 also shows the average cost, runtime, and iteration count for each method measured over 30 independent runs. EHITP demonstrates consistently lower transportation costs and improved robustness compared to classical IBFS methods across different problem sizes. Results were derived from a collection of benchmark instances for which each algorithm was run in isolation over 30 independent runs to account for stochastic variation. Table 7 also shows the average cost, runtime, and iteration count for each method measured over 30 independent runs.

Comparative performance

The convergence behavior of AML-FFA3 compared with the baseline algorithms, where faster descent and improved stability can be observed. We observe from the tabulated results ( Table 3) that EHITP was able to produce lower final transportation costs than any of the baseline IBFS methods on every instance. Comparatively, EHITP reduced this cost gap by more than 50% on average relative to the initial IBFS, regardless of the initial IBFS, while MODI rarely achieved comparable solution quality or robustness. Such and other related works suggest an emerging but less-active trend in customized IBFS heuristics and hybrid refinements, typically reporting better performance than NWC/LCM/VAM as well as, in some instances, approximate-optimal costs.

Among the medium-scale cases (3×5, 5×10 problems), EHITP achieved average costs that were 8–12% lower than those of the next-best heuristics. The standard deviation of more than 30 runs was also lower orders of magnitude, indicating not only a more stable solution, but also less sensitive to the initial solution.

Convergence behaviors

Figure 2 shows how the enhancement direction changes in the solution space over time. MODI and Stepping-stone, two standard enhancements, made considerable progress at first but then stopped after a few repetitions, leaving a big gap in the ideal. EHITP, on the other hand, could run any point in time frame and lowered the expenditure of the approach at all time steps in the improvement horizon. Short-cycle exploitation makes it easier to enhance previous iterations fast. Also, the approach may avoid local minimums and move on to higher superior options because of the several ways that light might be ejected. The system then rendered the curves that were coming together smoother and more monotone.

Validation by statistics

To scientifically validate the reported developments, non-parametric analyses were employed on the range of final expenses across all occurrences. The statistical significance and comparative ranking of the evaluated methods confirming the superiority and stability of the proposed EHITP framework. Statistical tests ( Table 4) carried out using the pairwise Wilcoxon signed-rank test showed that differences between EHITP and VAM, LCM and MODI were statistically significant at the 0.05 level. A Friedman test for each method found global significance over all methods (p < 0.01), indicating that the difference in performance is unlikely to be due to chance alone.²² Such improved results further substantiate that EHITP continues to maintain statistically proven superiority. Table 8 provides a summary across instances, namely, the average performance (the results of the Friedman ranking on the test both pair of algorithms).

Conclusion and future work

In this study, the Ester Hybrid Improvement Algorithm for the Transportation Problem (EHITP), a refinement framework designed to escape the stagnation typically found in classical post-optimization techniques such as MODI and Stepping-Stone, was proposed. Using three synergistic components $(i)$ focused exploitation using a guidance mechanism based on the MODI index for a local search methodology (to drive search towards promising regions), $(ii)$ short-cycle exploitation to increase number of search iterations within promising neighborhoods and $(iii)$ light ejection diversification moves (dedicated to local minima escape) EHITP obtained results consistently better than the (exhaustive) improvement heuristics.

Web-based experimental evaluations on benchmark instances of the transportation problem showed that EHITP was able to lower the final Cost of transportation while being more stable in repeated runs, indicating robustness with respect to initial conditions. For many test gaps, EHITP filled more than half of the convex hull distance between classical IBFS solutions (such as VAM, LCM) and the optimal (or near optimal) known solutions, while demanding only a modest additional computational expense. The trade-off between solution quality and efficiency suggests that EHITP will be a valuable tool for real-world application scenarios, where both cost minimization and computational tractability are crucial.

Future work

• Generalize EHITP to Multi-Objective Transportation Problems by considering Cost, time, and environmental emissions to be consistent with sustainable logistics-related objectives (e.g., sustainable hub location)
• Extend EHITP to stochastic and fuzzy transportation problems to make it more suitable for robust demand, supply or cost parameters uncertainty.
• Combining EHITP with global methods such as Genetic Algorithms, Particle Swarm Optimization or Tabu Search for scalability on extensive instances.
• Compose EHITP with fast network flow solvers (e.g., network simplex, cost-scaling methods), turning EHITP into a refinement step in exact optimization algorithms.

Data availability

Datasets: The complete datasets used in this study were fully simulated by the authors for experimental and methodological validation. None of the simulated data is based on actual observed records, images, or elements of real world or copyrighted datasets.

All the simulated datasets including the problem instances, the parameters the algorithms were run under, and the output final results are available open access in Zenodo: EHITP: Ester Hybrid Improvement Algorithm for the Transportation Problem. https://doi.org/10.5281/zenodo.17433753²³

Data are available under the terms of the Creative Commons Attribution 4.0 International license (CC-BY 4.0).

Software availability

• Source code: https://github.com/iraqt-alt/HMPCS-ML-Solar-Forecasting
• Archived software available from: https://doi.org/10.5281/zenodo.17559972

The Zenodo archive represents the official, citable version of the EHITP implementation and includes preprocessing scripts, optimization modules, simulation code, and experiment configuration files.

The software is released under the MIT License (OSI-approved) to ensure transparency, reproducibility, and unrestricted academic reuse.

A GitHub repository is maintained only as a development mirror and is not considered the primary archived reference.

Acknowledgment

The authors gratefully acknowledge the University of Fallujah for providing the facilities and financial assistance that enabled the completion of this study.

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Version 2

VERSION 2 PUBLISHED 14 Feb 2026

Author details Author details

Faten Hameed Sabty
Roles: Conceptualization, Supervision

Noor Hassan Ali
Roles: Data Curation, Formal Analysis, Writing – Original Draft Preparation

Iraq T. Abbas
Roles: Conceptualization, Methodology, Project Administration, Supervision, Writing – Review & Editing

Hanan Ali Cheachan
Roles: Data Curation, Investigation

Competing interests

No competing interests were disclosed.

Grant information

This research was financially supported by the University of Fallujah, Iraq, through its academic research funding program. The support covered data analysis, computational resources, and publication preparation.
The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Article Versions (2)

version 2

Revised

Published: 24 Apr 2026, 15:263

https://doi.org/10.12688/f1000research.172115.2

version 1

Published: 14 Feb 2026, 15:263

https://doi.org/10.12688/f1000research.172115.1

© 2026 Hameed Sabty F et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Hameed Sabty F, Hassan Ali N, Abbas IT and Ali Cheachan H. EHITP: Ester Hybrid Improvement Algorithm for the Transportation Problem [version 1; peer review: 1 approved, 1 not approved]. F1000Research 2026, 15:263 (https://doi.org/10.12688/f1000research.172115.1)

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Open Peer Review

Version 1

VERSION 1

PUBLISHED 14 Feb 2026

Views

Reviewer Report 06 Apr 2026

Annisa Kesy Garside, Universitas Muhammadiyah Malang, Malang, Indonesia

Not Approved

https://doi.org/10.5256/f1000research.189810.r469145

This manuscript introduces the Ester Hybrid Improvement (EHITP) algorithm which is designed to optimize the solution to the Transportation Problem (Transportation Problem - TP). The author aims to bridge the gap between the classic Initial Basic Feasible Solution (IBFS) method and the optimal solution through heuristic hybridization. This paper evaluates the performance of the algorithm compared to traditional methods such as the North-West Corner (NWC), Least Cost Method (LCM), and Vogel's Approximation Method (VAM) using small-scale numerical examples and statistical testing.
Reviewer's Main Comment
Although this topic is relevant in the field of operational research, the current version of the manuscript contains several fundamental weaknesses, critical inconsistencies, and technical inaccuracies that make it not scientifically eligible.
Specific Criticism and Necessary Revision

Critical Inconsistencies in Presentation and Accuracy This manuscript has a very serious internal contradiction, which shows a lack of precision in the compilation:

Algorithm Naming: In the results section, especially on Table 7 (page 9), the author presents the results for an algorithm named "HMPCS-ML", even though the entire paper discusses "EHITP".
Image Reference Error: In the methodology section (page 5), the text refers to Image 2 as "AML-FFA3 improvement pipeline", but the description (caption) on Image 2 is written "EHITP improvement pipeline".Recommendation: The author should conduct a comprehensive audit of the manuscript to ensure that the algorithm explained in the methodology is the same algorithm used in the experiment. All references to "HMPCS-ML" and "AML-FFA3" should be corrected or clarified.

Data Availability and Reproducibility

The "Data Availability" section includes a GitHub link that should contain the EHITP algorithm code. However, the repository points to a project called "HMPCS-ML-Solar-Forecasting".

Issue: The code in the repository is related to solar energy prediction and has no relation to the Transportation Problem or the EHITP algorithm discussed in this paper.
Recommendation: To comply with F1000Research's open data policy, the author must provide the correct source code link and include the specific dataset (matrix) used in this research

Statistical Analysis and Interpretation

Mathematically incorrect interpretation of statistical results:

Contradiction: In Table 5, the results of the Wilcoxon Signed-Rank test show that the p-value ranges from 0.5000 to 1.0000. Statistically, this value shows that there is no significant difference between EHITP and the comparative method. However, the authors conclude in the text that EHITP is "significantly superior."
Recommendation: The authors should reevaluate their statistical tests. If the improvement in results is not statistically significant, then claims of superiority should be reduced or supported by testing on a much larger and diverse dataset to find real significance.

Literature Review and State of the Art (SOTA)

This paper claims to provide modern improvements, but the majority of comparisons are made against very old methods (NWC, VAM, LCM).

Lack of Modern Context: To show real "improvements", the authors should compare their methods with contemporary hybrid algorithms or metaheuristics (for example, modified Genetic Algorithm or Particle Swarm Optimization for TP) published in the last 5 years.

Technical Details and Methodological

Methodology: Some of the functions mentioned in the algorithm (such as light_ejection_shake) do not provide sufficient mathematical or logical explanations, so that it is difficult for the reader to fully understand the "hybrid" mechanism.
Scalability: Experiments are limited to very small matrices. Authors should include at least one large-scale benchmark problem to prove the algorithm's efficiency in real-world scenarios.

Conclusion and Revision Mandatory Points
In order for this article to be considered scientifically worthy, the following points must be improved:

Fix all naming inconsistencies (EHITP vs HMPCS-ML vs AML-FFA3).
Provide a link to the correct GitHub repository containing the specific code for the EHITP algorithm and the dataset used.
Align conclusions with statistical data; if the p-value is high, the claim of "statistical significance" should be removed or the test should be extended.
Explain the details of the sub-functions of the algorithm to allow full transparency and replication by other researchers

Is the work clearly and accurately presented and does it cite the current literature?

No
Is the study design appropriate and is the work technically sound?

No
Are sufficient details of methods and analysis provided to allow replication by others?

No
If applicable, is the statistical analysis and its interpretation appropriate?

No
Are all the source data underlying the results available to ensure full reproducibility?

No
Are the conclusions drawn adequately supported by the results?

No

Competing Interests: No competing interests were disclosed.

Reviewer Expertise: Operations Research, Optimization Algorithms, Logistics, and Supply Chain Management.

I confirm that I have read this submission and believe that I have an appropriate level of expertise to state that I do not consider it to be of an acceptable scientific standard, for reasons outlined above.

CITE

Report a concern

Author Response 24 Apr 2026

iraq abbas, Mathematics, University of Baghdad Al-Jaderyia Campus College of Science, Baghdad, 00964, Iraq

24 Apr 2026

Author Response

Dear Editor and Reviewers,

Thank you for your valuable comments and constructive feedback.

We have carefully revised the manuscript to address all the concerns raised. In particular, we ... Continue reading Dear Editor and Reviewers,

Thank you for your valuable comments and constructive feedback.

We have carefully revised the manuscript to address all the concerns raised. In particular, we have corrected inconsistencies in the methodology and results sections, unified the terminology throughout the paper, and improved the statistical interpretation of the results. Additionally, the experimental tables and dataset descriptions have been updated to ensure clarity and consistency.

We believe that the revised version of the manuscript significantly improves its scientific quality and presentation.

We appreciate your time and consideration.

Sincerely,
Iraq T. Abbas
Dear Editor and Reviewers,

Thank you for your valuable comments and constructive feedback.

We have carefully revised the manuscript to address all the concerns raised. In particular, we have corrected inconsistencies in the methodology and results sections, unified the terminology throughout the paper, and improved the statistical interpretation of the results. Additionally, the experimental tables and dataset descriptions have been updated to ensure clarity and consistency.

We believe that the revised version of the manuscript significantly improves its scientific quality and presentation.

We appreciate your time and consideration.

Sincerely,
Iraq T. Abbas
Competing Interests: The authors declare that they have no competing interests. Close
Report a concern
Respond or Comment

COMMENTS ON THIS REPORT

Author Response 24 Apr 2026

iraq abbas, Mathematics, University of Baghdad Al-Jaderyia Campus College of Science, Baghdad, 00964, Iraq

24 Apr 2026

Author Response

Dear Editor and Reviewers,

Thank you for your valuable comments and constructive feedback.

We have carefully revised the manuscript to address all the concerns raised. In particular, we ... Continue reading Dear Editor and Reviewers,

Thank you for your valuable comments and constructive feedback.

We have carefully revised the manuscript to address all the concerns raised. In particular, we have corrected inconsistencies in the methodology and results sections, unified the terminology throughout the paper, and improved the statistical interpretation of the results. Additionally, the experimental tables and dataset descriptions have been updated to ensure clarity and consistency.

We believe that the revised version of the manuscript significantly improves its scientific quality and presentation.

We appreciate your time and consideration.

Sincerely,
Iraq T. Abbas
Dear Editor and Reviewers,

Thank you for your valuable comments and constructive feedback.

We have carefully revised the manuscript to address all the concerns raised. In particular, we have corrected inconsistencies in the methodology and results sections, unified the terminology throughout the paper, and improved the statistical interpretation of the results. Additionally, the experimental tables and dataset descriptions have been updated to ensure clarity and consistency.

We believe that the revised version of the manuscript significantly improves its scientific quality and presentation.

We appreciate your time and consideration.

Sincerely,
Iraq T. Abbas
Competing Interests: The authors declare that they have no competing interests. Close
Report a concern

Views

Reviewer Report 23 Feb 2026

Hussam Abid Ali Mohammed, University of Kerbala, Karbala, Karbala Governorate, Iraq

Approved

https://doi.org/10.5256/f1000research.189810.r459029

The study presents a robust hybrid framework Enhanced Heuristic for the Transportation Problem (EHITP) that effectively integrates adaptive IBFS initialization with MODI-guided local search and diversification strategies. It demonstrates improved solution quality, enhanced stability across runs, and statistically validated performance, ... Continue reading

Is the work clearly and accurately presented and does it cite the current literature?

Yes
Is the study design appropriate and is the work technically sound?

Yes
Are sufficient details of methods and analysis provided to allow replication by others?

Yes
If applicable, is the statistical analysis and its interpretation appropriate?

Partly
Are all the source data underlying the results available to ensure full reproducibility?

Yes
Are the conclusions drawn adequately supported by the results?

Yes

Competing Interests: No competing interests were disclosed.

Reviewer Expertise: Operation Research

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.

CITE

Report a concern

Author Response 27 Apr 2026

iraq abbas, Mathematics, University of Baghdad Al-Jaderyia Campus College of Science, Baghdad, 00964, Iraq

27 Apr 2026

Author Response

Dear Reviewer,
Thank you very much for your valuable and constructive feedback. We sincerely appreciate your positive evaluation of our work and your recognition of its contribution.
Regarding the statistical ... Continue reading Dear Reviewer,
Thank you very much for your valuable and constructive feedback. We sincerely appreciate your positive evaluation of our work and your recognition of its contribution.
Regarding the statistical analysis, we have carefully reviewed this aspect and made additional clarifications and minor improvements in the revised version to further strengthen the presentation and interpretation of the results.
We are grateful for your insightful comments, which helped us improve the quality of the manuscript.
Sincerely,
Iraq T. Abbas
Dear Reviewer,
Thank you very much for your valuable and constructive feedback. We sincerely appreciate your positive evaluation of our work and your recognition of its contribution.
Regarding the statistical analysis, we have carefully reviewed this aspect and made additional clarifications and minor improvements in the revised version to further strengthen the presentation and interpretation of the results.
We are grateful for your insightful comments, which helped us improve the quality of the manuscript.
Sincerely,
Iraq T. Abbas
Competing Interests: No competing interests were disclosed. Close
Report a concern
Respond or Comment

COMMENTS ON THIS REPORT

Author Response 27 Apr 2026

iraq abbas, Mathematics, University of Baghdad Al-Jaderyia Campus College of Science, Baghdad, 00964, Iraq

27 Apr 2026

Author Response

Dear Reviewer,
Thank you very much for your valuable and constructive feedback. We sincerely appreciate your positive evaluation of our work and your recognition of its contribution.
Regarding the statistical ... Continue reading Dear Reviewer,
Thank you very much for your valuable and constructive feedback. We sincerely appreciate your positive evaluation of our work and your recognition of its contribution.
Regarding the statistical analysis, we have carefully reviewed this aspect and made additional clarifications and minor improvements in the revised version to further strengthen the presentation and interpretation of the results.
We are grateful for your insightful comments, which helped us improve the quality of the manuscript.
Sincerely,
Iraq T. Abbas
Dear Reviewer,
Thank you very much for your valuable and constructive feedback. We sincerely appreciate your positive evaluation of our work and your recognition of its contribution.
Regarding the statistical analysis, we have carefully reviewed this aspect and made additional clarifications and minor improvements in the revised version to further strengthen the presentation and interpretation of the results.
We are grateful for your insightful comments, which helped us improve the quality of the manuscript.
Sincerely,
Iraq T. Abbas
Competing Interests: No competing interests were disclosed. Close
Report a concern

Comments on this article Comments (0)

Version 2

VERSION 2 PUBLISHED 14 Feb 2026

Open Peer Review

Reviewer Status

Reviewer Reports

	Invited Reviewers
	1	2
Version 2 (revision) 24 Apr 26
Version 1 14 Feb 26	read	read

Hussam Abid Ali Mohammed, University of Kerbala, Karbala, Iraq
Annisa Kesy Garside, Universitas Muhammadiyah Malang, Malang, Indonesia

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Reviewer Report

13 Views

06 Apr 2026 | for Version 1

Annisa Kesy Garside, Universitas Muhammadiyah Malang, Malang, Indonesia

13 Views Cite this report Responses(1)

Not Approved

Critical Inconsistencies in Presentation and Accuracy This manuscript has a very serious internal contradiction, which shows a lack of precision in the compilation:

Algorithm Naming: In the results section, especially on Table 7 (page 9), the author presents the results for an algorithm named "HMPCS-ML", even though the entire paper discusses "EHITP".
Image Reference Error: In the methodology section (page 5), the text refers to Image 2 as "AML-FFA3 improvement pipeline", but the description (caption) on Image 2 is written "EHITP improvement pipeline".Recommendation: The author should conduct a comprehensive audit of the manuscript to ensure that the algorithm explained in the methodology is the same algorithm used in the experiment. All references to "HMPCS-ML" and "AML-FFA3" should be corrected or clarified.

Data Availability and Reproducibility

The "Data Availability" section includes a GitHub link that should contain the EHITP algorithm code. However, the repository points to a project called "HMPCS-ML-Solar-Forecasting".

Issue: The code in the repository is related to solar energy prediction and has no relation to the Transportation Problem or the EHITP algorithm discussed in this paper.
Recommendation: To comply with F1000Research's open data policy, the author must provide the correct source code link and include the specific dataset (matrix) used in this research

Statistical Analysis and Interpretation

Mathematically incorrect interpretation of statistical results:

Contradiction: In Table 5, the results of the Wilcoxon Signed-Rank test show that the p-value ranges from 0.5000 to 1.0000. Statistically, this value shows that there is no significant difference between EHITP and the comparative method. However, the authors conclude in the text that EHITP is "significantly superior."
Recommendation: The authors should reevaluate their statistical tests. If the improvement in results is not statistically significant, then claims of superiority should be reduced or supported by testing on a much larger and diverse dataset to find real significance.

Literature Review and State of the Art (SOTA)

This paper claims to provide modern improvements, but the majority of comparisons are made against very old methods (NWC, VAM, LCM).

Lack of Modern Context: To show real "improvements", the authors should compare their methods with contemporary hybrid algorithms or metaheuristics (for example, modified Genetic Algorithm or Particle Swarm Optimization for TP) published in the last 5 years.

Technical Details and Methodological

Methodology: Some of the functions mentioned in the algorithm (such as light_ejection_shake) do not provide sufficient mathematical or logical explanations, so that it is difficult for the reader to fully understand the "hybrid" mechanism.
Scalability: Experiments are limited to very small matrices. Authors should include at least one large-scale benchmark problem to prove the algorithm's efficiency in real-world scenarios.

Conclusion and Revision Mandatory Points
In order for this article to be considered scientifically worthy, the following points must be improved:

Fix all naming inconsistencies (EHITP vs HMPCS-ML vs AML-FFA3).
Provide a link to the correct GitHub repository containing the specific code for the EHITP algorithm and the dataset used.
Align conclusions with statistical data; if the p-value is high, the claim of "statistical significance" should be removed or the test should be extended.
Explain the details of the sub-functions of the algorithm to allow full transparency and replication by other researchers

Is the work clearly and accurately presented and does it cite the current literature?

No
Is the study design appropriate and is the work technically sound?

No
Are sufficient details of methods and analysis provided to allow replication by others?

No
If applicable, is the statistical analysis and its interpretation appropriate?

No
Are all the source data underlying the results available to ensure full reproducibility?

No
Are the conclusions drawn adequately supported by the results?

No

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

Operations Research, Optimization Algorithms, Logistics, and Supply Chain Management.

Respond to this report

Responses (1)

Author Response

24 Apr 2026

iraq abbas, Mathematics, University of Baghdad Al-Jaderyia Campus College of Science, Baghdad, 00964, Iraq

Dear Editor and Reviewers,

Thank you for your valuable comments and constructive feedback.

We have carefully revised the manuscript to address all the concerns raised. In particular, we have corrected inconsistencies in the methodology and results sections, unified the terminology throughout the paper, and improved the statistical interpretation of the results. Additionally, the experimental tables and dataset descriptions have been updated to ensure clarity and consistency.

We believe that the revised version of the manuscript significantly improves its scientific quality and presentation.

We appreciate your time and consideration.

Sincerely,
Iraq T. Abbas

View more View less

Competing Interests

The authors declare that they have no competing interests.

Back to all reports

Reviewer Report

10 Views

23 Feb 2026 | for Version 1

Hussam Abid Ali Mohammed, University of Kerbala, Karbala, Karbala Governorate, Iraq

10 Views Cite this report Responses(1)

Approved

Is the work clearly and accurately presented and does it cite the current literature?

Yes
Is the study design appropriate and is the work technically sound?

Yes
Are sufficient details of methods and analysis provided to allow replication by others?

Yes
If applicable, is the statistical analysis and its interpretation appropriate?

Partly
Are all the source data underlying the results available to ensure full reproducibility?

Yes
Are the conclusions drawn adequately supported by the results?

Yes

Competing Interests

No competing interests were disclosed.

Reviewer Expertise

Operation Research

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.

Respond to this report

Responses (1)

Author Response

27 Apr 2026

iraq abbas, Mathematics, University of Baghdad Al-Jaderyia Campus College of Science, Baghdad, 00964, Iraq

Dear Reviewer,
Thank you very much for your valuable and constructive feedback. We sincerely appreciate your positive evaluation of our work and your recognition of its contribution.
Regarding the statistical analysis, we have carefully reviewed this aspect and made additional clarifications and minor improvements in the revised version to further strengthen the presentation and interpretation of the results.
We are grateful for your insightful comments, which helped us improve the quality of the manuscript.
Sincerely,
Iraq T. Abbas

View more View less

Competing Interests

No competing interests were disclosed.

Alongside their report, reviewers assign a status to the article:

Approved - the paper is scientifically sound in its current form and only minor, if any, improvements are suggested

Approved with reservations - A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit.

Not approved - fundamental flaws in the paper seriously undermine the findings and conclusions

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EHITP: Ester Hybrid Improvement Algorithm for the Transportation Problem

Abstract

Background

Methods

Results

Conclusions

Keywords

Introduction

Table 1. Selected recent IBFS/Improvement methods (2020–2025).

Illustrative figures

Figure 1. EATI initialization pipeline.

Figure 2. EHITP improvement pipeline.

Expanded discussion: Positioning EATI and EHITP

Suggested experiments and reporting

Proposed method: EHITP

Methodology EATI and EHITP

Overview

EATI – Step-by-step algorithm

EHITP – Step-by-step algorithm

Datasets and experimental design

Metrics and statistics

Table 2. Dataset summary.

Table 3. Per-Instance results (Mean over runs).

Table 4. Ablation study.

Table 5. Statistical tests.

Table 6. Dataset summary.

Table 7. Per-instance results (averaged over 30 runs).

Table 8. Statistical summary across instances.

Reproducibility

Experimental setup

Results and discussion

Comparative performance

Convergence behaviors

Validation by statistics

Conclusion and future work

Future work

Data availability

Software availability

Acknowledgment

References

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