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 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.

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.

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.

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 ← X 0 ; bestCost ← cost ( X ) ; stall ← 0

2: for iter = 1 . . maxIter do

3:  ( U , V ) ← solve _ potentials _ from _ basis ( X )

4:  Δ ← C − ( U ⊕ V )

5:  if all Δ _ ij ≥ 0 then

6:   X ← light _ ejection _ shake ( X , C )

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

8:  else

9:   S ← k best cells by ( − Δ _ ij ) , preferring short cycles

10:   cycle ∗ ← argmax gain from cycles in S

11:   X ← augment _ along ( cycle ∗ )

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

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 ≥ 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 ≥ 0

Cycle Improvement: θ = min { x kl | ( k , l ) in cycle with ′−′ }

Stopping Conditions •

No improvement: Z k = Z k − 1

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).

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 ≥ 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.

• 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.

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

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).

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.

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.

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.

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).

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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)