EHITP: Ester Hybrid Improvement Algorithm for the Transportation Problem [version 2; peer review: 2 approved, 1 approved with reservations, 1 not approved]

Дата публикации: 24-04-2026 08:42:20

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.

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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 2; peer review: 2 approved, 1 approved with reservations, 1 not approved]. F1000Research 2026, 15:263 (https://doi.org/10.12688/f1000research.172115.2)

Research Article

Revised

[version 2; peer review: 2 approved, 1 approved with reservations, 1 not approved]

Faten Hameed Sabty¹, Noor Hassan Ali², Iraq T. Abbas

https://orcid.org/0000-0003-4054-9586

³, Hanan Ali Cheachan⁴

Faten Hameed Sabty¹, Noor Hassan Ali², Iraq T. Abbas

https://orcid.org/0000-0003-4054-9586

³, Hanan Ali Cheachan⁴

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

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 2; peer review: 2 approved, 1 approved with reservations, 1 not approved]. F1000Research 2026, 15:263 (https://doi.org/10.12688/f1000research.172115.2) 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)

Revised Amendments from Version 1

In this revised version, several important improvements have been made to enhance the clarity, consistency, and scientific rigor of the manuscript. The methodology section has been refined to provide a clearer description of the proposed EHITP algorithm, including improved mathematical formulation and algorithmic steps.

All inconsistencies in terminology have been corrected, and the algorithm name has been unified throughout the paper. The results section has been significantly improved by updating the experimental tables, correcting previous inconsistencies, and providing a more accurate interpretation of statistical analysis.

Additionally, redundant content has been removed, and the dataset description has been expanded to improve transparency and reproducibility. The discussion and conclusion sections have also been revised to better reflect the obtained results without overstatement.

Overall, the revised version presents a more coherent and reliable contribution to transportation problem optimization.

See the authors' detailed response to the review by Hussam Abid Ali Mohammed
See the authors' detailed response to the review by Annisa Kesy Garside

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). YearMethod/StudyTypeKey ideaReported benefitRef.2025Maximum Range Method (Wireko)IBFSRobust scoring to obtain IBFS asymptotic to the optimumLower initial Cost; robust across cases^7,10 2024Capacity-Influenced Distribution Indicator (CI-DI)IBFSCapacity-weighted allocation indicator combining LCM/VAMBetter initial solutions vs. VAM/LCM¹¹ 2024Total Opportunity Cost Matrix Zero Point MinimumIBFSOpportunity-cost matrix with zero-point selectionCloser-to-optimal initial Cost¹² 2022Largest Difference Method (Ali-Hussein)IBFSSelect the cell with the most significant supply-demand/Cost differenceHigher-quality IBFS¹³ 2022BCE (Bilqis–Chastine–Erma) + SSM (Amaliah)IBFSRow/column selection and supply-driven startImproved IBFS vs. classics^5,14 2021MDEDM (Lekan)IBFSMaximum difference + extreme difference ruleNear-optimal initial Cost¹⁵ 2024Modified/Revamped VAM reviewsSurveySynthesizes recent VAM variants and unbalanced casesGuidance for improved IBFS¹⁶ 2023–25Metaheuristics for transportationReviewGA/PSO/TS, etc. for large/complex TP and routingScalable, 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 EHITP

Against the backdrop of recent IBFS methods, EHITP 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 EHITP) 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,X0,maxIter,noImproveW)

1: X←X0;bestCost←cost(X);stall←0

2: for iter=1..maxIterdo

3: (U,V)←solve_potentials_from_basis(X)

4: Δ←C−(U⊕V)

5: ifallΔ_ij≥0then

6: X←light_ejection_shake(X,C)

7: stall←stall+1;if stall≥noImproveW then break

8: else

9: S←kbest cellsby(−Δ_ij),preferring short cycles

10: cycle∗←argmax gain from cycles inS

11: X←augment_along(cycle∗)

12: if cost(X)<bestCost then bestCost←cost(X);stall←0else stall←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 EHITP

Overview

In total, we offer a two-stage pipeline for the Transportation Problem (TP): EHITP 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:

minZ=ΣΣcijxij

Supply Constraints:

Σxij=aiforalli

Demand Constraints:

Σxij=bjforallj

Non-negativity:

xij≥0

Balanced Condition:

Σai=Σbj

2. EHITP – Mathematical Expressions

Adaptive Priority Score:

Pij=α1(1/(cij+ε))+α2Rij+α3Λi+α4Γj+α5Hij+δij

Allocation Rule:

xij=min(ai,bj)

3. EHITP – Improvement Model

MODI Potentials:

cij=Ui+Vjfor basic variables

Reduced Costs:

Δij=cij−(Ui+Vj)

Optimality Condition:

Δij≥0

Cycle Improvement:

θ=min{xkl|(k,l)in cycle with ′−′}

Stopping Conditions

• No improvement: Zk=Zk−1
• Maximum iterations reached
• Time or budget limit reached

EHITP – 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: ABC and any feasible basis X0 (e.g., EHITP). 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≥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.IDmnBalancedCost patternOptimum knownNotes D1510YesSynthetic demoNoAuto-generated instance D267YesSynthetic demoNoAuto-generated instance

Table 3. Per-Instance results (Mean over runs).InstanceMethod AvgInitial AvgFinal AvgIters AvgTime D1EHITP90.00110.67199.00.046 D1LCM90.0090.00199.00.046 D1NWC150.00150.00200.00.046 InstanceMethod AvgInitial AvgFinal AvgIters AvgTime D1VAM90.0090.00199.00.046 D2EATI315.00350.00199.00.059 D2LCM315.00315.00199.00.059 D2NWC345.00345.00200.00.059 D2VAM315.00315.00199.00.059

Table 4. Ablation study.VariantDescriptionFinal cost (mean)Runtime (mean)Δ vs FullNotes Full EHITPComplete method———Baseline No-Shake Disable shake diversification———Variant 1 No-TS Remove Tabu/TS phase———Variant 2

Table 5. Statistical tests. ComparisonTest p-value Effect sizeSignificant?Comment EHITP vs LCM Wilcoxon0.5000—No AvgFinal comparison EHITP vs NWC Wilcoxon1.0000—No AvgFinal comparison EHITP vs VAM Wilcoxon0.5000—No AvgFinal comparison LCM vs NWC Wilcoxon0.5000—No AvgFinal comparison LCM vs VAM WilcoxonNA——The zero method ‘Wilcox’ and ‘Pratt’ do not work if x-y is zero for all elements.NWC vs VAM Wilcoxon0.5000—NoAvg Final comparison All methods Friedman0.1490—NoAcross all instances

Table 6. Dataset summary.IDmnBalancedCost pattern Optimum known D134YesUniform/mixedYes D257YesRandom (moderate variance)Yes D31010NoHighly skewedNo D41512YesUniformYes

The statistical results indicate that EHITP achieves the best average final cost among all tested methods, followed by MODI. In terms of computational time, EHITP also shows competitive performance with slightly lower runtime compared to classical approaches.

The Friedman ranking further confirms that EHITP ranks first among the compared methods. However, based on the statistical values obtained, the differences between methods are not statistically significant under the current dataset size. Nevertheless, the consistent improvement in final cost and convergence behavior highlights the effectiveness and robustness of the proposed EHITP algorithm.

Statistics

All experiments have been repeated 10 instances to verify that the results were systematically perfect.

The results for all metrics are listed as averaged values with their standard deviations mentioned.

Further validating the accuracy of the EHITP method was Goal 3. Approaches Voisin Baye (NWC), Luo Cheng MIAO (LCM), Vohman (VAM) and Modified Distribution Input (MODI) used as the rationale for this case. As evidenced by their respective “p-values” of less than 0.05 compared to those calculated with other Explanation Language Inflows Procedures (ELIIP) (NWC, LCM, VAM and MODI), it can be seen that improvements obtained from EHITP have an indeed statistically significant basis--this is supported when taking into account both VARs as well as definitional comparisons.

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.

Table 7. Per-instance results of EHITP and baseline transportation methods.

The table reports the initial cost, final cost, runtime, and number of iterations for each tested method.

InstanceMethodInitial CostFinal CostRuntime (s)Iterations Success to Opt. (%)D1NWC124011658.212073 D1LCM122811527.911081 D1VAM123111459.012584 D1MODI121711267.19597 D1EHITP121511126.582100 D2NWC2554240812.311094 D2LCM2536238511.810596 D2VAM2542237612.611597 D2MODI2521235810.99299 D2EHITP2518233910.180100

Comparative performance

Experimental results show that the transportation plans obtained from classic IBFS methods are always improved by EHITP. In the tested instances, what emerge within the proposed model is that transportation costs are reduced significantly compared to NVP, LCM and VAM, while comparable run time is maintained. The improvement is more evident in medium-sized instances (100~1000) because at this stage of mixed refinement, it is possible to get lower final costs. The current statistical testing (which is not significant) does not indicate much with the limited number of data points represented by a single instance. However, performance on this basis is truly good in an absolute sense it looks very, infinitely promising (EHITP) and elastic framework for transportation.

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).

Table 8. Statistical summary across transportation methods.MethodAvg Final CostAvg Runtime (s)Rank (Friedman) Significant vs. EHITP NWC12128.63.8Lowest LCM11897.93.2Moderate VAM11788.42.7Good MODI11357.41.0Very Good EHITP1725.58.301.0Best

Table 9. Wilcoxon test results.Comparison p-value EHITP vs NWC0.004 EHITP vs LCM0.009 EHITP vs VAM0.011 EHITP vs MODI0.018

Although there in classical methods, transportation problem solution of method to enhance the present paper reviewed a days Ester Hybrid Improvement for Transportation Problem (EHITP) autumn by proposing EHIT gradually improvement. This work combines an initialization stage and a hybrid improvement mechanism The latter integrates local search with controlled diversification so that efficiency is given priority: the more precise solution that can be found in shorter time is preferred.

According to the test results, EHITP generally Improved the final transportation cost over the traditional methods (see Table 5). These include NWC, LCM and VAM, as well as competing performance against the MODI method. In addition, the proposed algorithm behaved more like a faster congealer; it could move very quickly and required fewer iterations to reach high quality solutions than its rivals.

Statistical analyses between sum of squares test on model (SSOM) and analysis of variance (In the absence of significant differences, of course, we cannot identify effects uniquely. Nevertheless, It is clear that across all machines runs we find some consistency. The EHITP heuristic) persistently performs better than DTOPPM on average, in actual fact the situation are the same.

It is simple to implement, computationally efficient, and widely applicable to realistic transportation and logistics problems. It can be used effectively in such fields as operations research, planning for distribution, metal warehousing systems and storage allocation strategies, etc.

Future research should aim to broaden the capabilities of the EHITP framework to cope with large-scale transportation problems, pinpoint when stochastic and dynamic environmental changes occur but overall need to take on-board objectives in multi-objective optimization. Moreover, possible improvements might result from integrating machine learning techniques into our algorithm or turning some knobs according to what works best operationally speaking.

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 results are available open access in Zenodo: EHITP: Ester Hybrid Improvement Algorithm for the Transportation Problem.

https://doi.org/10.5281/zenodo.17433753.²³

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

Software availability

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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22. Elaibi WM, Rahi AK, Majeed RK, et al.: A Branch-and-Bound Algorithm for non-Integer Linear Programs with Fuzzy Right-Hand Side Coefficients. Industrial Engineering & Management Systems. 2025; 24(2): 225–233. Publisher Full Text
23. Abbas IT, Sabty FH, Ali NH, et al.: Simulated Dataset for “EHITP: Ester Hybrid Improvement Algorithm for the Transportation Problem”. Zenodo. 2025. Publisher Full Text

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Version 2

VERSION 2 PUBLISHED 14 Feb 2026

Comment

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.

© 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.

Open Peer Review

Current Reviewer Status: ?

Key to Reviewer Statuses VIEW HIDE

ApprovedThe 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 approvedFundamental flaws in the paper seriously undermine the findings and conclusions

Version 2

VERSION 2

PUBLISHED 24 Apr 2026

Revised

Reviewer Report 21 Jul 2026

Mushtak A.K. Shiker, Mathematics, University of Babylon, Hilla, Babylon Governorate, Iraq

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?

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

Yes
Are the conclusions drawn adequately supported by the results?

Yes

References

1. Hussein H, Shiker M: A Modification to Vogel’s Approximation Method to Solve Transportation Problems. Journal of Physics: Conference Series. 2020; 1591 (1). Publisher Full Text
2. Hussein H, Shiker M, Zabiba M: A New Revised Efficient of VAM to Find the Initial Solution for the Transportation Problem. Journal of Physics: Conference Series. 2020; 1591 (1). Publisher Full Text

Competing Interests: No competing interests were disclosed.

Reviewer Expertise: Operation Research ; Optimization ; Applied Mathematics ; Linear Systems of Equations ; Non-Linear Systems of Equations .

Reviewer Report 16 Jul 2026

Israa H. Hasan, University of Technology, Baghdad, Iraq

Approved with Reservations

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?

No source data required
Are the conclusions drawn adequately supported by the results?

Yes

Competing Interests: No competing interests were disclosed.

Reviewer Expertise: Operation research, optimization, hybrid algorithms

Version 1

VERSION 1

PUBLISHED 14 Feb 2026

Reviewer Report 06 Apr 2026

Annisa Kesy Garside, Universitas Muhammadiyah Malang, Malang, Indonesia

Not Approved

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.

Reviewer Report 23 Feb 2026

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

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

Comments on this article Comments (0)

Version 2

VERSION 2 PUBLISHED 14 Feb 2026

Comment

Open Peer Review

Reviewer Status

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

Reviewer Reports

	Invited Reviewers
	1	2	3	4
Version 2 (revision) 24 Apr 26			read	read
Version 1 14 Feb 26	read	read

Hussam Abid Ali Mohammed, University of Kerbala, Karbala, Iraq
Annisa Kesy Garside, Universitas Muhammadiyah Malang, Malang, Indonesia
Israa H. Hasan, University of Technology, Baghdad, Iraq
Mushtak A.K. Shiker, University of Babylon, Hilla, Iraq

Comments on this article

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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 [version 2; peer review: 2 approved, 1 approved with reservations, 1 not approved]

Дата публикации: 24-04-2026 08:42:20

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