A New Algorithm for Finding Initial Basic Feasible Solutions of Transportation Problems

Abstract

This study introduces a deterministic fractional-penalty refinement of Vogel’s Approximation Method (VAM) for generating high-quality initial basic feasible solutions (IBFS) in classical transportation problems. Unlike the traditional additive regret measure employed in VAM, the proposed method uses a multiplicative contrast ratio between the two smallest admissible costs in each row and column. This modification preserves the allocation structure of VAM while introducing scale-invariant prioritization that improves sensitivity to relative cost differences.The method was evaluated on thirty-four benchmark transportation problems drawn from the literature and self-constructed large-scale instances (up to $10\times 20$). Performance was assessed using percentage optimality gaps relative to optimal solutions obtained via the Stepping–Stone and MODI procedures. Across all instances, the proposed approach achieved a mean optimality gap of $2.78\%$, compared to $5.22\%$ for classical VAM, $14.97\%$ for the Least Cost Method (LCM), and $45.78\%$ for the Northwest Corner Method (NWCM). Dispersion of deviations was also reduced, indicating improved robustness across heterogeneous cost structures Statistical validation confirms the improvement over VAM: the paired t-test yielded $t=-3.17$ ($p=0.00163$, one-sided), and the Wilcoxon signed-rank test produced $p=6.10\times {10}^{-5}$. Computational experiments further show that the refinement does not increase runtime relative to classical IBFS procedures.The proposed method therefore constitutes a structured enhancement of VAM that improves initial solution quality while maintaining computational simplicity.

1. Introduction

The transportation problem (TP) is one of the earliest and most studied linear programming models. It was originally formulated by Hitchcock and later extended within the framework of linear programming [1,2]. Constructive heuristics for generating an initial basic feasible solution (IBFS) remain important because the quality of the IBFS can influence the number of iterations required in subsequent optimality procedures such as the Stepping-Stone or MODI methods [3].

Classical IBFS procedures include the North-West Corner Method (NWCM), the Least Cost Method (LCM), and Vogel’s Approximation Method (VAM). Among these, VAM is widely regarded as the most cost-sensitive because it incorporates a penalty defined as the difference between the two smallest admissible costs in each row or column [4]. An effective heuristic approach for determining an initial basic feasible solution to the transportation problem has been proposed, offering a practical efficiency and solution quality [5]. Despite its effectiveness, the additive penalty used in VAM is sensitive to absolute cost magnitudes and scaling. In heterogeneous cost environments, proportional dispersion between competing allocation options may not be fully captured by simple cost differences. This observation motivates the investigation of alternative penalty formulations that preserve the constructive structure of VAM while redefining its prioritization metric.

This study proposes a ratio-based (fractional) penalty defined as the quotient of the second-smallest and smallest admissible costs in each active line. The resulting measure emphasizes proportional cost contrast and remains invariant under positive cost scaling. Importantly, the allocation structure of VAM is retained; only the penalty computation is modified.

The objective of this paper is therefore not to introduce an entirely new heuristic family, but rather to provide a mathematically transparent and computationally reproducible refinement of the VAM penalty structure. Extensive benchmark testing and statistical validation are conducted to evaluate performance consistency across diverse problem instances.

The contribution of this study lies in introducing a scale-invariant multiplicative penalty mechanism within the classical Vogel’s Approximation Method framework. Unlike the traditional additive regret measure used in VAM, the proposed fractional penalty captures proportional dispersion between competing transportation costs. This modification preserves the constructive allocation structure of VAM while providing an alternative prioritization mechanism that can improve the quality and stability of initial basic feasible solutions. The effectiveness of the approach is validated through computational experiments on thirty-four benchmark transportation problems together with statistical analysis of optimality gaps.

Research Gap and Contribution

Although numerous heuristics exist for obtaining initial basic feasible solutions (IBFS) in transportation problems, many widely used procedures remain rooted in either minimum-cost selection or additive regret ideas. Common IBFS methods include the Row Minimum Method, the Column Minimum Method, and the Maximum Difference Method. Other classical baselines are the Least Cost Method (LCM) and Vogel’s Approximation Method (VAM).

Recent studies have introduced alternative penalty models and hybrid allocation strategies for constructing IBFS. For example, the Maximum Range Method was proposed to improve allocation prioritization [6]. Alternative procedures for generating initial feasible solutions have also been explored [7]. Furthermore, new penalty-based models motivated by probabilistic or distributional mechanisms have been developed [8]. Other algorithmic strategies for determining IBFS have also been reported in recent studies [9]. However, the theoretical implications of multiplicative cost dispersion in row/column prioritization remain largely unexplored.

Classical VAM employs an additive regret defined as the difference between the two smallest admissible costs. While effective, this measure is not scale-invariant and may underrepresent proportional dispersion when costs vary significantly across rows.

This study introduces a deterministic fractional-penalty refinement that replaces additive regret with a multiplicative contrast ratio. The contribution lies not in altering the allocation structure of VAM, but in modifying the prioritization mechanism in a theoretically motivated and statistically validated manner.

2. Literature Review

Transportation problems remain a central topic in operations research due to their wide applicability in logistics, supply chain management, and production planning. Classical constructive heuristics for generating an initial basic feasible solution (IBFS), namely the North-West Corner Method (NWCM), the Least Cost Method (LCM), and Vogel’s Approximation Method (VAM), are extensively discussed in standard operations research textbooks [10,11,12].

The continued relevance of these classical heuristics has been demonstrated in several comparative studies. Performance evaluations of traditional IBFS methods confirm their effectiveness as baseline techniques for transportation problems [13]. Further investigations into their computational behavior show that these heuristics remain competitive for small- and medium-scale transportation instances [14]. Additional empirical studies provide supporting evidence of their applicability in practical scenarios [15,16]. These classical methods also serve as important benchmarks for evaluating newly proposed algorithms [17].

The North-West Corner Method (NWCM) is attractive because of its simplicity and low computational cost. However, it ignores transportation costs and allocates shipments solely by traversing the transportation tableau from the top-left corner, which often leads to relatively coarse initial solutions [18]. The Least Cost Method (LCM) improves upon NWCM by selecting the smallest available unit transportation cost at each allocation step. This strategy generally produces better initial solutions than NWCM [19]. However, LCM may suffer from myopic allocation decisions and sensitivity to tie-breaking rules when multiple cells have identical costs [20].

Vogel’s Approximation Method (VAM) introduces a penalty-based strategy defined by the difference between the two smallest costs in each row or column. This penalty mechanism allows VAM to prioritize allocations where suboptimal decisions would be most costly [21]. Several studies have shown that VAM typically produces solutions closer to optimality than NWCM and LCM [22].

In practice, classical IBFS procedures must also address implementation issues such as balancing through dummy rows or columns, degeneracy prevention, and deterministic tie-breaking rules. Improper handling of these issues can significantly affect solution quality and reproducibility [23]. Standard operations research references emphasize the need for systematic implementation rules to ensure consistent computational performance [24]. The transportation problem has been extensively studied in both applied and theoretical contexts, with contributions focusing on cost minimization and optimization techniques [25,26]. However, existing approaches still face challenges in improving computational efficiency during optimality checks. These limitations motivate ongoing research aimed at developing improved IBFS algorithms with stronger cost-awareness and enhanced computational robustness.

3. Methodology: Finding Initial Basic Feasible Solutions

3.1. Notation

Let $C=\left(c_{ij}\right)$ be the $m\times n$ cost matrix with supplies $a_i>0$ ($i=1,\dots ,m$) and demands $b_j>0$ ($j=1,\dots ,n$). A transportation problem (TP) is balanced if ${\sum }_{i=1}^m{a}_i={\sum }_{j=1}^n{b}_j$. If it is unbalanced, we add a dummy column (when total supply exceeds total demand) or a dummy row (when total demand exceeds total supply). The dummy line has cost 0 to all of its cells, and the added supply/demand equals the absolute imbalance.

Throughout all methods:

• Stopping criterion: Continue allocating until all supplies and demands are satisfied.
• Degeneracy safeguard: An IBFS must contain exactly $m+n-1$ basic cells. If a row and a column are both exhausted by the same allocation, cross out one line and leave the other active with a zero residual, or place a tiny bookkeeping allocation $\epsilon$ in a zero-cost eligible cell (do not alter totals) to maintain $m+n-1$ basic variables.
• Tie-breaking (deterministic): When ties occur, use: (i) prefer row-first over column when methods require choosing a line; (ii) within a line, choose the lowest cost; if still tied, choose the leftmost (smallest column index); if still tied, choose the topmost (smallest row index). Using a fixed rule ensures reproducibility.
• Quality measure: The IBFS cost is

  $$Z_{\mathrm{IBFS}}=\sum _{i=1}^m\sum _{j=1}^n{c}_{ij}{x}_{ij}$$

  (1)

3.2. General Steps for Finding IBFS

1.

Balance the problem. If total supply and total demand differ, add a zero-cost dummy row (when demand exceeds supply) or dummy column (when supply exceeds demand) so totals match.

2.

Initialize. Mark all rows and columns as active, with their remaining supply/demand.

3.

Choose a line or cell (selection rule). Apply one of the rules below to decide where to allocate next.

4.

Allocate. In the chosen cell $(i,j)$, set

$$x_{ij}=min\{\mathrm{remaining}\mathrm{supply}\mathrm{of}\mathrm{row}i,\mathrm{remaining}\mathrm{demand}\mathrm{of}\mathrm{column}j\}$$

(2)

5.

Update the rim. Reduce the row’s remaining supply and the column’s remaining demand by $x_{ij}$.

6.

Cross out satisfied lines. Any row/column that reaches zero becomes inactive.

7.

Handle degeneracy. If a single allocation makes both a row and a column reach zero, cross out one and keep the other active with zero balance (or place a formal $\epsilon$ in an eligible cell) so that the final IBFS has exactly $m+n-1$ basic cells.

8.

Repeat. Recompute the quantities required by the selection rule on the reduced tableau and return to Step 3 until all supplies and demands are satisfied.

9.

Stop. When all rim requirements are met, the current $x_{ij}$ constitute an IBFS.

10.

The IBFS cost is computed using Equation (1).

Selection Rules (To Use in Step 3)

• North–West Corner Method (NWCM): Start at the top-left active corner, allocate, then move right when a column is satisfied and down when a row is satisfied.
• Least Cost Method (LCM): Choose the globally cheapest active cell.
• Vogel’s Approximation Method (VAM): For each active line, the penalty is

  $$P=c_{\left(2\right)}-c_{\left(1\right)}$$

  (3)

  where $c_{\left(1\right)}$ and $c_{\left(2\right)}$ are the smallest and second smallest costs in that line.
• Mean Minima Method: For each active line, compute

  $$M={\displaystyle \frac{c_{\left(1\right)}+c_{\left(2\right)}}{2}}$$

  (4)

  and select the line with the smallest mean.

3.3. Computational Complexity Analysis

At each allocation step, the algorithm requires identifying the two smallest elements in each active row and column. This operation has linear complexity in the number of active cells per iteration. Since exactly $m+n-1$ allocations are performed in constructing an IBFS, the overall worst-case computational complexity remains comparable to classical VAM, namely $\mathcal{O}\left(mn\right)$ per iteration and

$$\mathcal{O}\left(mn\right(m+n\left)\right).$$

(5)

Because the allocation structure is identical to VAM and only the penalty computation differs, the proposed algorithm does not introduce additional asymptotic computational burden.

4. Results and Discussions

4.1. The New Algorithm for Finding Initial Basic Feasible Solutions (IBFS) of Transportation Problems

The following are the steps involved in the new algorithm.

4.1.1. Step 1: Balance Check

If the total supply does not equal total demand, the problem must be balanced. This condition is expressed as

$$\sum _{i=1}^m{a}_i=\sum _{j=1}^n{b}_j$$

(6)

If Equation (6) does not hold, a dummy row (for excess demand) or dummy column (for excess supply) with zero transportation costs is introduced to balance the problem.

4.1.2. Step 2: Row and Column Fractional Penalties

For each active row i, determine the two smallest costs among the admissible cells. These are defined as

$$m_i^{\left(1\right)}=\underset{j}{min}{c}_{ij}$$

(7)

and

$$m_i^{\left(2\right)}=\text{second-smallest}\mathrm{of}\left\{c_{ij}\right\}$$

(8)

The fractional penalty for row i is then defined as

$$p_i={\displaystyle \frac{m_i^{\left(2\right)}}{m_i^{\left(1\right)}}}$$

(9)

Similarly, for each active column j, the two smallest costs are obtained and the column penalty is defined as

$$q_j={\displaystyle \frac{m_j^{\left(2\right)}}{m_j^{\left(1\right)}}}$$

(10)

If $m^{\left(1\right)}=0$, the corresponding line is treated as having very large priority to avoid division by zero.

4.1.3. Step 3: Selection and Allocation

The row or column with the largest penalty is selected according to

$$P^{\ast }=max\{p_i,q_j\}$$

(11)

Within the selected row or column, the minimum-cost cell $(i^{\ast },j^{\ast })$ is chosen and the allocation is computed as

$$x_{i^{\ast }{j}^{\ast }}=min(a_{i^{\ast }},b_{j^{\ast }})$$

(12)

The supply and demand values are then updated accordingly.

4.1.4. Step 4: Iteration

After each allocation, the penalties are recomputed for the reduced tableau and Steps 2–3 are repeated until all supplies and demands are satisfied.

4.1.5. Step 5: IBFS Cost

The total cost of the obtained initial basic feasible solution is computed as

$$Z_{\mathrm{IBFS}}=\sum _{i=1}^m\sum _{j=1}^n{c}_{ij}{x}_{ij}$$

(13)

4.2. Monotonicity Property of the Fractional Penalty

The ratio-based penalty also exhibits a monotonicity property that clarifies its prioritization behavior relative to the smallest admissible cost in a row or column.

Lemma 1. 

Let $m^{\left(1\right)}$ and $m^{\left(2\right)}$ denote the smallest and second-smallest admissible costs in an active row (or column) with $m^{\left(1\right)}>0$ and $m^{\left(2\right)}\ge m^{\left(1\right)}$. Define the fractional penalty

$$p={\displaystyle \frac{m^{\left(2\right)}}{m^{\left(1\right)}}}.$$

(14)

Then the penalty p is strictly increasing in $m^{\left(2\right)}$ and strictly decreasing in $m^{\left(1\right)}$.

Proof.  

Consider the penalty function $p(m^{\left(1\right)},m^{\left(2\right)})={\displaystyle \frac{m^{\left(2\right)}}{m^{\left(1\right)}}}$ for $m^{\left(1\right)}>0$. The partial derivatives are

$${\displaystyle \frac{\partial p}{\partial m^{\left(2\right)}}}={\displaystyle \frac{1}{m^{\left(1\right)}}}>0,$$

(15)

and

$${\displaystyle \frac{\partial p}{\partial m^{\left(1\right)}}}=-{\displaystyle \frac{m^{\left(2\right)}}{{\left(m^{\left(1\right)}\right)}^2}}<0.$$

(16)

Thus, increasing the second-smallest admissible cost increases the penalty, while increasing the smallest admissible cost decreases the penalty. Consequently, rows or columns with stronger proportional cost contrast receive higher priority in the allocation process.    □

This monotonicity property provides a theoretical justification for the prioritization mechanism of the fractional penalty. In contrast to the additive penalty used in classical Vogel’s Approximation Method, the ratio-based formulation emphasizes proportional cost dispersion, thereby capturing relative dominance among competing allocation options.

4.3. Demonstration of the New Algorithm

To illustrate the procedure, we consider a transportation problem adapted from standard literature [27]. The cost matrix, supplies, and demands are given in

Table 1. Transportation Tableau (BUA Cement Company Data).

	D1	D2	D3	D4	D5	D6	Supply
P1	25	30	20	40	45	37	37
P2	30	25	20	30	40	20	22
P3	40	20	40	35	45	22	32
P4	25	24	50	27	30	25	14
Demand	15	20	15	25	20	10	105

.

Since total supply equals total demand (105), the problem is balanced.

4.3.1. Iteration 1

The computed row and column penalties for the first iteration are presented in

Table 2. Iteration 1: Penalty Workout (Rows and Columns).

Line	${\mathit{m}}^{\left(1\right)}$	${\mathit{m}}^{\left(2\right)}$	Penalty
Row P1	20	25	$p_1=25/20=1.25$
Row P2	20	20	$p_2=20/20=1.00$
Row P3	20	22	$p_3=22/20=1.10$
Row P4	24	25	$p_4=25/24\approx 1.04$
Col D1	25	25	$q_1=25/25=1.00$
Col D2	20	24	$q_2=24/20=1.20$
Col D3	20	20	$q_3=20/20=1.00$
Col D4	27	30	$q_4=30/27\approx 1.11$
Col D5	30	40	$q_5=40/30\approx 1.33$
Col D6	20	22	$q_6=22/20=1.10$

.

$$p_i={\displaystyle \frac{m_i^{\left(2\right)}}{m_i^{\left(1\right)}}},q_j={\displaystyle \frac{m_j^{\left(2\right)}}{m_j^{\left(1\right)}}}$$

(17)

The maximum penalty occurs in column $D_5$. The minimum cost in column $D_5$ is $c_{45}=30$.

$$x_{45}=min(14,20)=14$$

(18)

Row $P_4$ is exhausted; demand of $D_5$ reduces to 6.

4.3.2. Iteration 2

Recompute penalties on the reduced tableau.

A tie occurs between row $P_1$ and column $D_2$. Using the deterministic row-first rule, select row $P_1$. Minimum cost in row $P_1$ is at $D_3$:

$$x_{13}=min(37,15)=15$$

(19)

Column $D_3$ is satisfied; supply of $P_1$ reduces to 22.

4.3.3. Iteration 3

Row $P_2$ has the largest penalty. Minimum cost in row $P_2$ is at $D_6$.

$$x_{26}=min(22,10)=10$$

(20)

Column $D_6$ is satisfied; supply of $P_2$ reduces to 12.

4.3.4. Iteration 4

Row $P_3$ has the largest penalty. Minimum cost in row $P_3$ is at $D_2$:

$$x_{32}=min(32,20)=20$$

(21)

Column $D_2$ is satisfied; supply of $P_3$ reduces to 12.

4.3.5. Iteration 5

Row $P_1$ now has the largest penalty. Minimum cost in row $P_1$ is at $D_1$:

$$x_{11}=min(22,15)=15$$

(22)

Column $D_1$ is satisfied; supply of $P_1$ reduces to 7.

4.3.6. Iteration 6

Row $P_2$ has the largest penalty. Minimum cost in row $P_2$ is at $D_4$:

$$x_{24}=min(12,25)=12$$

(23)

Row $P_2$ is exhausted; demand of $D_4$ reduces to 13.

4.3.7. Iteration 7

Row $P_3$ has the largest penalty. Minimum cost in row $P_3$ is at $D_4$:

$$x_{34}=min(12,13)=12$$

(24)

Row $P_3$ is exhausted; demand of $D_4$ reduces to 1.

4.3.8. Iteration 8

Only row $P_1$ remains active.

$$x_{14}=min(7,1)=1$$

(25)

Column $D_4$ is satisfied; supply of $P_1$ reduces to 6.

4.3.9. Iteration 9

Final allocation:

$$x_{15}=6$$

(26)

All supplies and demands are satisfied.

Final Initial Basic Feasible Solution

$$X=\left(\begin{array}{cccccc}15& 0& 15& 1& 6& 0\\ 0& 0& 0& 12& 0& 10\\ 0& 20& 0& 12& 0& 0\\ 0& 0& 0& 0& 14& 0\end{array}\right)$$

(27)

Number of allocations:

$$m+n-1=4+6-1=9$$

(28)

Hence, the solution is non-degenerate.

IBFS Cost

$$\begin{array}{c}\hfill Z_{\mathrm{IBFS}}=25\left(15\right)+20\left(15\right)+40\left(1\right)+45\left(6\right)+30\left(12\right)+20\left(10\right)+20\left(20\right)+35\left(12\right)+30\left(14\right)\end{array}$$

(29)

Thus, the fractional-penalty algorithm produces an initial solution with total transportation cost:

$$\begin{array}{|c|}\hline Z_{\mathrm{IBFS}}=2785.\\ \hline\end{array}$$

(30)

To further substantiate performance consistency, we computed the mean, median, and standard deviation of optimality gaps across all benchmark instances. Additionally, paired comparisons between VAM and the new method were conducted to determine whether improvements were systematic.

4.4. Optimality Gap Analysis

To evaluate the practical relevance of the new method, we compare the IBFS costs produced by each heuristic against the known optimal solution cost $Z^{\ast }$ obtained via standard optimality procedures.

For each benchmark instance, we compute the percentage optimality gap as:

$$\mathrm{Gap}(\%)={\displaystyle \frac{Z_{\mathrm{IBFS}}-Z^{\ast }}{Z^{\ast }}}\times 100$$

(31)

This measure provides a normalized assessment of solution quality independent of cost magnitude. Lower values indicate closer proximity to optimality.

Across the benchmark set, the fractional-penalty variant frequently exhibits equal or smaller gaps compared to VAM, LCM, and NWCM. While improvements are moderate in magnitude for some instances, they are consistent across several categories of cost structures.

4.5. Comparison of the New Algorithm with Existing Algorithms in Terms of Solutions

Table 3. Presents Comparison of the New Algorithm with the Existing Algorithms in Terms of their Solutions (Examples 1–4).

Example	Problem Size	Sources and Problems	VAM	NWCM	LCM	New Algorithm
1	$5\times 5$	[28] $C_{ij}$ = [4 3 1 2 6; 5 2 3 4 5; 3 5 6 3 2; 2 4 4 5 3; 4 3 6 5 1]; $S_i$ = [65 50 40 20 25]; $D_j$ = [60 60 30 40 10].	420	760	420	420
2	$4\times 4$	[29] $C_{ij}$ = [20 16 14 20; 9 15 16 10; 8 13 5 9; 9 6 5 11]; $S_i$ = [9 8 7 5]; $D_j$ = [5 10 5 9].	308	392	308	308
3	$6\times 6$	[29] $C_{ij}$ = [5 1 2 3 4 7; 7 2 3 1 5 6; 9 1 9 5 2 3; 6 5 8 4 1 4; 8 7 11 6 4 5; 2 5 7 5 2 1]; $S_i$ = [400 500 300 150 600 350]; $D_j$ = [300 500 700 300 250 250].	8400	9600	8600	8400
4	$3\times 4$	[30] $C_{ij}$ = [7 3 8 2; 5 6 11 12; 10 4 7 6]; $S_i$ = [100 200 300]; $D_j$ = [80 170 190 160].	3210	4010	4010	3210

,

Table 4. Presents Comparison of the New Algorithm with the Existing Algorithms in Terms of their Solutions (Examples 5–12).

Example	Problem Size	Sources and Problems	VAM	NWCM	LCM	New Algorithm
5	$5\times 5$	[31] $C_{ij}$ = [46 74 9 28 99; 12 75 6 36 48; 35 199 4 5 71; 61 81 44 88 9; 85 60 14 25 79]; $S_i$ = [461 277 356 488 393]; $D_j$ = [278 60 461 116 1060].	64,499	68,969	72,174	64,499
6	$3\times 3$	[32] $C_{ij}$ = [6 8 10; 7 11 11; 4 5 12]; $S_i$ = [150 175 275]; $D_j$ = [200 100 300].	5125	5925	4550	4525
7	$4\times 6$	[25] $C_{ij}$ = [9 12 9 6 9 10; 7 3 7 7 5 5; 6 5 9 11 3 11; 6 8 11 2 2 10]; $S_i$ = [5 6 2 9]; $D_j$ = [4 4 6 2 4 2].	112	139	114	112
8	$4\times 5$	[26] $C_{ij}$ = [4 3 1 2 6; 5 2 3 4 5; 3 5 6 3 2; 2 4 4 5 3]; $S_i$ = [80 60 40 20]; $D_j$ = [60 60 30 40 10].	450	670	420	420
9	$4\times 4$	[11] $C_{ij}$ = [10 30 25 15; 20 15 20 10; 10 30 20 20; 30 40 35 45]; $S_i$ = [14 10 15 12]; $D_j$ = [10 15 12 15].	1005	1220	1075	1000
10	$4\times 6$	[30] $C_{ij}$ = [1 2 1 4 5 2; 3 3 2 1 4 3; 4 2 5 9 6 2; 3 1 7 3 4 6]; $S_i$ = [30 50 75 20]; $D_j$ = [20 40 30 10 50 25].	450	740	450	450
11	$3\times 3$	[31] $C_{ij}$ = [11 9 6; 12 14 11; 10 8 10]; $S_i$ = [40 50 40]; $D_j$ = [55 45 30].	1200	1490	1200	1200
12	$4\times 5$	[7]. $C_{ij}$ = [30 50 40 60 35; 65 35 45 30 25; 35 40 60 40 30; 20 30 50 45 35]; $S_i$ = [20 15 25 20]; $D_j$ = [15 18 10 17 20].	2600	3105	2675	2600

,

Table 5. Presents Comparison of the New Algorithm with the Existing Algorithms in Terms of their Solutions (Examples 13–18).

Example	Problem Size	Sources and Problems	VAM	NWCM	LCM	New Algorithm
13	$3\times 4$	[17] $C_{ij}$ = [15 12 10 8; 17 18 21 14; 14 15 10 21]; $S_i$ = [24 8 18]; $D_j$ = [11 9 21 9].	571	760	595	571
14	$3\times 5$	[32] $C_{ij}$ = [5 8 6 6 3; 4 7 7 6 5; 8 4 6 6 4]; $S_i$ = [800 500 900]; $D_j$ = [400 400 500 400 800].	9800	13,100	10,200	9200
15	$3\times 4$	[32] $C_{ij}$ = [3 1 7 4; 2 6 5 9; 8 3 3 2]; $S_i$ = [300 400 500]; $D_j$ = [250 350 400 200].	2850	4400	2900	2850
16	$3\times 5$	[26] $C_{ij}$ = [4 1 2 4 4; 2 3 2 2 2; 3 5 2 4 4]; $S_i$ = [60 35 40]; $D_j$ = [22 45 20 18 30].	275	363	305	273
17	$2\times 7$	[12] $C_{ij}$ = [5 19 12 70 66 74 283; 103 89 81 26 23 62 97]; $S_i$ = [4000 47,700]; $D_j$ = [21,600 15,600 15,600 19,500 16,800 10,500 8100].	2,332,700	2,336,000	2,348,300	1,992,700
18	$3\times 4$	[23] $C_{ij}$ = [3 48 14 2; 4 2 30 10; 36 8 12 12]; $S_i$ = [24 24 2]; $D_j$ = [6 12 3 44].	224	906	308	180

,

Table 6. Presents Comparison of the New Algorithm with the Existing Algorithms in Terms of their Solutions (Examples 19–25).

Example	Problem Size	Sources and Problems	VAM	NWCM	LCM	New Algorithm
19	$4\times 6$	[21] $C_{ij}$ = [9 12 9 6 9 10; 7 3 7 7 5 5; 6 5 9 11 3 11; 6 8 11 2 2 10]; $S_i$ = [5 6 2 9]; $D_j$ = [4 4 6 2 4 2].	112	119	114	112
20	$3\times 4$	[21] $C_{ij}$ = [3 5 7 6; 2 5 8 2; 3 6 9 2]; $S_i$ = [50 75 25]; $D_j$ = [20 20 50 60].	650	670	650	650
21	$4\times 5$	[20] $C_{ij}$ = [10 8 9 5 13; 7 9 8 10 4; 9 3 7 10 6; 11 4 8 3 9]; $S_i$ = [100 80 70 90]; $D_j$ = [60 40 100 50 90].	2130	3010	2070	2070
22	$4\times 6$	[8] $C_{ij}$ = [9 12 9 6 9 10; 7 3 7 7 5 5; 6 5 9 11 3 11; 6 8 11 2 2 10]; $S_i$ = [2 5 6 9]; $D_j$ = [2 2 4 4 4 6].	109	167	122	109
23	$4\times 3$	[22] $C_{ij}$ = [3 4 6; 7 3 8; 6 4 5; 7 5 2]; $S_i$ = [100 80 90 120]; $D_j$ = [110 110 60].	840	1010	1210	840
24	$5\times 4$	[16] $C_{ij}$ = [60 120 75 180; 58 100 60 165; 62 110 65 170; 65 115 80 175; 70 135 85 195]; $S_i$ = [8000 9200 6250 4900 6100]; $D_j$ = [5000 2000 10,000 6000].	2,164,000	2,398,000	2,383,000	2,160,000

,

Table 7. Presents Comparison of the New Algorithm with the Existing Algorithms in Terms of their Solutions (Examples 25–31).

Example	Problem Size	Sources and Problems	VAM	NWCM	LCM	New Algorithm
25	$3\times 5$	[19] $C_{ij}$ = [4 1 3 4 4; 2 3 2 2 3; 3 5 2 4 4]; $S_i$ = [60 35 40]; $D_j$ = [22 45 20 18 30].	305	363	307	290
26	$3\times 3$	[19] $C_{ij}$ = [6 4 1; 3 8 7; 4 4 2]; $S_i$ = [50 40 60]; $D_j$ = [20 95 35].	555	710	555	555
27	$5\times 5$	[24] $C_{ij}$ = [73 40 9 79 20; 62 93 96 8 13; 96 65 80 50 65; 57 58 29 12 87; 56 23 87 18 12]; $S_i$ = [8 7 9 3 5]; $D_j$ = [6 8 10 4 4].	1128	1994	1123	1104
28	$4\times 6$	[26] $C_{ij}$ = [25 30 20 40 45 37; 30 25 20 30 40 20; 40 20 40 35 45 22; 25 24 50 27 30 25]; $S_i$ = [37 22 32 14]; $D_j$ = [15 20 15 25 20 10].	2850	3195	2878	2785
29	$3\times 4$	[10] $C_{ij}$ = [8 6 10 9; 9 12 13 7; 14 9 16 5]; $S_i$ = [35 50 40]; $D_j$ = [45 20 30 30].	1020	1180	1080	1020
30	$2\times 3$	[6] $C_{ij}$ = [7 8 10; 9 7 8]; $S_i$ = [50 50]; $D_j$ = [40 40 40].	730	730	800	730
31	$5\times 10$	Self. $C_{ij}$ = [6 9 5 7 8 6 4 9 3 7; 7 5 8 6 9 7 5 8 6 4; 8 6 9 7 5 8 6 9 7 5; 5 8 6 9 7 5 8 6 9 7; 9 7 5 8 6 9 7 5 8 6]. $S_i$ = [35 50 45 40 30]. $D_j$ = [15 20 25 10 30 18 12 22 20 28].	973	1434	973	973

and

Table 8. Presents Comparison of the New Algorithm with the Existing Algorithms in Terms of their Solutions (Examples 32–34).

Example	Problem Size	Sources and Problems	VAM	NWCM	LCM	New Algorithm
32	$10\times 10$	Self. $C_{ij}$ = [4 6 9 7 8 5 7 6 8 9; 7 4 6 9 7 8 5 7 6 8; 8 7 4 6 9 7 8 5 7 6; 6 8 7 4 6 9 7 8 5 7; 5 6 8 7 4 6 9 7 8 5; 9 5 6 8 7 4 6 9 7 8; 7 9 5 6 8 7 4 6 9 7; 6 7 9 5 6 8 7 4 6 9; 8 6 7 9 5 6 8 7 4 6; 6 8 6 7 9 5 6 8 7 4]. $S_i$ = [30 25 40 35 20 50 25 30 20 25]. $D_j$ = [35 20 25 30 15 40 25 35 30 45].	1285	1590	1285	1285
33	$10\times 15$	Self. $C_{ij}$ = [4 7 5 8 6 4 7 5 8 6 4 7 5 8 6; 6 4 7 5 8 6 4 7 5 8 6 4 7 5 8; 8 6 4 7 5 8 6 4 7 5 8 6 4 7 5; 5 8 6 4 7 5 8 6 4 7 5 8 6 4 7; 7 5 8 6 4 7 5 8 6 4 7 5 8 6 4; 9 7 5 8 6 9 7 5 8 6 9 7 5 8 6; 6 9 7 5 8 6 9 7 5 8 6 9 7 5 8; 8 6 9 7 5 8 6 9 7 5 8 6 9 7 5; 5 8 6 9 7 5 8 6 9 7 5 8 6 9 7; 7 5 8 6 9 7 5 8 6 9 7 5 8 6 9]. $S_i$ = [28 42 33 47 22 36 28 50 44 50]. $D_j$ = [22 27 32 36 18 42 30 24 28 20 26 38 34 28 15].	1819	2383	1819	1763
34	$10\times 20$	Self. $C_{ij}$ = [6 9 4 7 5 8 6 5 7 9 4 8 6 7 5 9 6 4 8 5; 7 6 8 5 9 4 7 6 8 5 9 6 5 8 7 4 6 9 5 7; 5 8 6 9 4 7 5 8 6 7 5 9 4 6 8 5 7 4 6 9; 8 5 7 6 9 5 8 4 6 9 7 5 8 6 4 7 5 8 6 7; 9 4 6 8 5 7 3 6 9 5 8 4 7 6 5 8 5 8 6 7; 4 7 5 8 6 9 7 5 8 6 9 7 4 6 9 5 8 7 5 6; 6 5 9 4 7 6 8 5 7 4 6 8 7 5 9 6 4 7 5 8; 5 7 4 6 9 7 5 7 4 6 9 7 5 7 4 6 9 7 5 7; 7 4 6 9 5 8 6 9 5 7 6 4 8 5 7 6 9 5 8 6; 6 8 5 7 4 6 9 7 5 7 4 6 9 7 5 7 4 6 9 7]. $S_i$ = [48 37 55 62 41 59 46 68 44 50]. $D_j$ = [22 22 24 28 26 20 34 25 21 27 23 29 35 23 24 26 28 24 24 25].	2146	3434	2235	2130

compare the initial basic feasible solution costs produced by the new algorithm with those obtained by VAM, NWCM, and LCM across the benchmark instances.

4.6. Statistical Validation of Comparative Results

To determine whether observed performance differences are systematic rather than incidental, aggregated statistics were computed across all benchmark instances. These include the mean, median, maximum, and standard deviation of optimality gaps for each heuristic.

Additionally, paired statistical tests were conducted between VAM and the proposed method. Both a paired t-test and a Wilcoxon signed-rank test were applied to the set of instance-wise gap differences.

The results indicate that the fractional-penalty variant achieves statistically significant improvements over classical VAM at conventional confidence levels in several benchmark subsets. However, in certain cases performance remains equivalent, confirming that the proposed method should be interpreted as a structured refinement rather than a universal dominance mechanism.

Reference Optimal Values

For each instance, optimality gaps were computed relative to the best-known optimal solution obtained via New Optimal Method or Stepping-Stone or MODI procedures. In cases where identical values were produced by multiple methods, the minimum verified solution value was used as the reference benchmark.

4.7. Interpretation of Aggregated Optimality Gap Statistics

Table 9. Aggregated Optimality Gap Statistics Across 34 Instances.

Method	Mean Gap (%)	Median Gap (%)	Std. Dev. (%)	Max Gap (%)
NWCM	45.78	24.89	73.05	403.33
LCM	14.97	2.57	29.07	71.11
VAM	5.22	0.00	7.91	42.50
Proposed Variant	2.78	0.00	5.89	31.25

reports aggregated optimality-gap statistics computed across the thirty-four benchmark instances. For each instance, the optimality gap is defined by Equation (28). where $Z^{\ast }$ denotes the optimal transportation cost obtained by the optimality procedures (Stepping–Stone/MODI).

The Northwest Corner Method (NWCM) produces the largest deviations from optimality, with a mean gap of $45.78\%$ and a maximum gap of $403.33\%$. This confirms the well-known limitation of NWCM as a feasibility-driven rule that does not exploit cost information.

The Least Cost Method (LCM) improves substantially over NWCM, reducing the mean gap to $14.97\%$ and the median to $2.57\%$. However, its dispersion remains relatively high (standard deviation $29.07\%$), indicating sensitivity to problem structure and allocation order.

Vogel’s Approximation Method (VAM) further tightens the initial solution quality, achieving a mean gap of $5.22\%$ and a median gap of $0.00\%$. The zero median indicates that at least half of the instances produce VAM initial costs equal to the optimal value.

The proposed fractional-penalty variant yields the smallest overall deviation, with a mean gap of $2.78\%$, median gap of $0.00\%$, and a lower variability (standard deviation $5.89\%$) than the other heuristics. This suggests that the proposed penalty refinement improves both average performance and stability across heterogeneous cost structures.

Although the maximum gap for the proposed method reaches $31.25\%$ on the most challenging instance, the aggregated statistics consistently indicate closer-to-optimal initial solutions than NWCM, LCM, and classical VAM under the same benchmark set.

4.8. Wilcoxon Signed-Rank Test Formulation

To assess whether the fractional-penalty variant produces systematically smaller optimality gaps than classical VAM, a paired Wilcoxon signed-rank test is conducted on the instance-wise optimality gaps.

Let

$$d_i={\mathrm{Gap}}_{\mathrm{Proposed},i}-{\mathrm{Gap}}_{\mathrm{VAM},i},i=1,2,\dots ,n$$

(32)

with $n=34$ denote the number of benchmark instances. Differences with $d_i=0$ are discarded, leaving $n_0$ nonzero paired differences.

Rank the absolute nonzero differences $|d_i|$ in ascending order and denote the rank of $|d_i|$ by $R_i$ (average ranks are used in case of ties). Define

$$W^{+}=\sum _{d_i>0}{R}_i,W^{-}=\sum _{d_i<0}{R}_i$$

(33)

The Wilcoxon signed-rank test statistic is commonly taken as

$$T=min(W^{+},W^{-})$$

(34)

For large samples, a normal approximation can be used:

$$Z={\displaystyle \frac{W^{+}-{\mu }_W}{{\sigma }_W}}$$

(35)

where

$${\mu }_W={\displaystyle \frac{n_0(n_0+1)}{4}},{\sigma }_W=\sqrt{{\displaystyle \frac{n_0(n_0+1)(2n_0+1)}{24}}}$$

(36)

The hypotheses are

$$H_0:\mathrm{median}\left(d_i\right)=0\mathrm{vs}.H_1:\mathrm{median}\left(d_i\right)<0$$

(37)

where $H_1$ corresponds to the new method having smaller gaps than VAM.

Using the 34 benchmark instances, the Wilcoxon signed-rank test (one-sided; Proposed < VAM) yielded $p=6.10\times {10}^{-5}$, indicating a statistically significant reduction in optimality gaps for the new method.

4.9. Paired t-Test Formulation

To further evaluate whether the proposed fractional-penalty variant yields smaller optimality gaps than classical VAM, a paired t-test is conducted.

Let

$$d_i={\mathrm{Gap}}_{\mathrm{Proposed},i}-{\mathrm{Gap}}_{\mathrm{VAM},i},i=1,2,\dots ,n$$

(38)

where $n=34$. The sample mean and standard deviation of differences are;

$$\overline{d}={\displaystyle \frac{1}{n}}\sum _{i=1}^n{d}_i$$

(39)

and the sample standard deviation of the paired differences is

$$s_d=\sqrt{{\displaystyle \frac{1}{n-1}}\sum _{i=1}^n{(d_i-\overline{d})}^2}$$

(40)

The paired t statistic is

$$t={\displaystyle \frac{\overline{d}}{s_d/\sqrt{n}}}$$

(41)

Under the null hypothesis

$$H_0:{\mu }_d=0$$

(42)

The statistic follows a t distribution with $n-1$ degrees of freedom. The directional alternative hypothesis is

$$H_1:{\mu }_d<0$$

(43)

corresponding to the new method having smaller gaps than VAM.

Using the 34 benchmark instances, the paired differences in optimality gaps yielded $\overline{d}=-2.44\%$ with standard deviation $s_d=4.49\%$. The resulting statistic was $t=-3.17$ with 33 degrees of freedom. The one-sided p-value (Proposed < VAM) was $p=0.00163$ (two-sided $p=0.00327$), indicating a statistically significant improvement at the 5% level.

Normality of paired differences was inspected prior to the t-test; therefore, both parametric and non-parametric tests were reported.

4.10. Graphical Comparison of Optimality Gaps

To further illustrate the comparative performance of the methods, we present a graphical representation of the optimality gaps across all benchmark instances as shown in Figure 1.

The mean optimality gaps for each heuristic are presented in Figure 2.

4.11. Benchmark Characteristics

The benchmark dataset comprises thirty-four transportation problems drawn from the literature and supplemented with self-generated cases. Problem sizes range from small (3 × 3) to medium and larger instances (up to 10 × 20).

Cost matrices include diverse structural patterns, including:

• Uniformly distributed costs,
• Highly skewed cost distributions,
• Clustered near-equal costs,
• Instances containing zero or near-zero costs.

This diversity enables evaluation of how multiplicative penalties behave under varying dispersion conditions.

4.12. Comparison of the Computational Speed of the New Algorithm with VAM

The New Algorithm has been compared with the Vogel’s Approximation Method (VAM). All computational experiments were conducted using MATLAB R2022b on a computer equipped with an Intel Core i7 processor (3.20 GHz), 16 GB RAM, running Windows 10 (64-bit). An elapsed time counter was incorporated into the implementation. Each algorithm was executed five times per dataset, and the average execution time was recorded to enhance reproducibility.

Table 10. Presents Comparison of the Computational Speed of the New Algorithm with VAM (Examples 1–6).

Example	Problem Size	Sources and Problems	VAM (Time in Seconds)	New Algorithm (Time in Seconds)
1	5 × 5	[28] $C_{ij}$ = [4 3 1 2 6; 5 2 3 4 5; 3 5 6 3 2; 2 4 4 5 3; 4 3 6 5 1] $S_i$ = [65 50 40 20 25] $D_j$ = [60 60 30 40 10]	0.040992	0.038122
2	4 × 4	[29] $C_{ij}$ = [20 16 14 20; 9 15 16 10; 8 13 5 9; 9 6 5 11] $S_i$ = [9 8 7 5] $D_j$ = [5 10 5 9]	0.060489	0.055432
3	6 × 6	[29] $C_{ij}$ = [5 1 2 3 4 7; 7 2 3 1 5 6; 9 1 9 5 2 3; 6 5 8 4 1 4; 8 7 11 6 4 5; 2 5 7 5 2 1] $S_i$ = [400 500 300 150 600 350] $D_j$ = [300 500 700 300 250 250]	0.071573	0.051505
4	3 × 4	[30] $C_{ij}$ = [7 3 8 2; 5 6 11 12; 10 4 7 6] $S_i$ = [100 200 300] $D_j$ = [80 170 190 160]	0.048811	0.045997
5	5 × 5	[31] $C_{ij}$ = [46 74 9 28 99; 12 75 6 36 48; 35 199 4 5 71; 61 81 44 88 9; 85 60 14 25 79] $S_i$ = [461 277 356 488 393] $D_j$ = [278 60 461 116 1060]	0.072355	0.054465
6	3 × 3	[32] $C_{ij}$ = [6 8 10; 7 11 11; 4 5 12] $S_i$ = [150 175 275] $D_j$ = [200 100 300]	0.066016	0.051732

,

Table 11. Presents Comparison of the Computational Speed of the New Algorithm with VAM (Examples 7–14).

Example	Problem Size	Sources and Problems	VAM (Time in Seconds)	New Algorithm (Time in Seconds)
7	4 × 6	[25] $C_{ij}$ = [9 12 9 6 9 10; 7 3 7 7 5 5; 6 5 9 11 3 11; 6 8 11 2 2 10] $S_i$ = [5 6 2 9] $D_j$ = [4 4 6 2 4 2]	0.053182	0.048898
8	4 × 5	[26] $C_{ij}$ = [4 3 1 2 6; 5 2 3 4 5; 3 5 6 3 2; 2 4 4 5 3] $S_i$ = [80 60 40 20] $D_j$ = [60 60 30 40 10]	0.045334	0.041321
9	4 × 4	[11] $C_{ij}$ = [10 30 25 15; 20 15 20 10; 10 30 20 20; 30 40 35 45] $S_i$ = [14 10 15 12] $D_j$ = [10 15 12 15]	0.063444	0.057456
10	4 × 6	[30] $C_{ij}$ = [1 2 1 4 5 2; 3 3 2 1 4 3; 4 2 5 9 6 2; 3 1 7 3 4 6] $S_i$ = [30 50 75 20] $D_j$ = [20 40 30 10 50 25]	0.052848	0.050397
11	3 × 3	[31] $C_{ij}$ = [11 9 6; 12 14 11; 10 8 10] $S_i$ = [40 50 40] $D_j$ = [55 45 30]	0.055022	0.052754
12	4 × 5	[7] $C_{ij}$ = [30 50 40 60 35; 65 35 45 30 25; 35 40 60 40 30; 20 30 50 45 35] $S_i$ = [20 15 25 20] $D_j$ = [15 18 10 17 20]	0.066100	0.066043
13	3 × 4	[17] $C_{ij}$ = [15 12 10 8; 17 18 21 14; 14 15 10 21] $S_i$ = [24 8 18] $D_j$ = [11 9 21 9]	0.061222	0.060881
14	3 × 5	[32] $C_{ij}$ = [5 8 6 6 3; 4 7 7 6 5; 8 4 6 6 4] $S_i$ = [800 500 900] $D_j$ = [400 400 500 400 800]	0.071123	0.068456

,

Table 12. Presents Comparison of the Computational Speed of the New Algorithm with VAM (Examples 15–23).

Example	Problem Size	Sources and Problems	VAM (Time in Seconds)	New Algorithm (Time in Seconds)
15	3 × 4	[32] $C_{ij}$ = [3 1 7 4; 2 6 5 9; 8 3 3 2] $S_i$ = [300 400 500] $D_j$ = [250 350 400 200]	0.028123	0.020764
16	3 × 5	[26] $C_{ij}$ = [4 1 2 4 4; 2 3 2 2 2; 3 5 2 4 4] $S_i$ = [60 35 40] $D_j$ = [22 45 20 18 30]	0.049365	0.049004
17	2 × 7	[12] $C_{ij}$ = [5 19 12 70 66 74 283; 103 89 81 26 23 62 97] $S_i$ = [4000 47,700] $D_j$ = [21,600 15,600 15,600 19,500 16,800 10,500 8100]	0.059800	0.055632
18	3 × 4	[23] $C_{ij}$ = [3 48 14 2; 4 2 30 10; 36 8 12 12] $S_i$ = [24 24 2] $D_j$ = [6 12 3 44]	0.067088	0.051764
19	4 × 6	[21] $C_{ij}$ = [9 12 9 6 9 10; 7 3 7 7 5 5; 6 5 9 11 3 11; 6 8 11 2 2 10] $S_i$ = [5 6 2 9] $D_j$ = [4 4 6 2 4 2]	0.053237	0.055003
20	3 × 4	[21] $C_{ij}$ = [3 5 7 6; 2 5 8 2; 3 6 9 2] $S_i$ = [50 75 25] $D_j$ = [20 20 50 60]	0.048811	0.045997
21	4 × 5	[20] $C_{ij}$ = [10 8 9 5 13; 7 9 8 10 4; 9 3 7 10 6; 11 4 8 3 9] $S_i$ = [100 80 70 90] $D_j$ = [60 40 100 50 90]	0.047493	0.046144
22	4 × 6	[8] $C_{ij}$ = [9 12 9 6 9 10; 7 3 7 7 5 5; 6 5 9 11 3 11; 6 8 11 2 2 10] $S_i$ = [2 5 6 9] $D_j$ = [2 2 4 4 4 6]	0.054543	0.054323
23	4 × 3	[22] $C_{ij}$ = [3 4 6; 7 3 8; 6 4 5; 7 5 2] $S_i$ = [100 80 90 120] $D_j$ = [110 110 60]	0.051605	0.038987

,

Table 13. Presents Comparison of the Computational Speed of the New Algorithm with VAM (Examples 24–30).

Example	Problem Size	Sources and Problems	VAM (Time in Seconds)	New Algorithm (Time in Seconds)
24	4 × 5	[16] $C_{ij}$ = [60 120 75 180; 58 100 60 165; 62 110 65 170; 65 115 80 175; 70 135 85 195] $S_i$ = [8000 9200 6250 4900 6100] $D_j$ = [5000 2000 10,000 6000]	0.051246	0.048190
25	3 × 5	[19] $C_{ij}$ = [4 1 3 4 4; 2 3 2 2 3; 3 5 2 4 4] $S_i$ = [60 35 40] $D_j$ = [22 45 20 18 30]	0.056765	0.054321
26	3 × 3	[19] $C_{ij}$ = [6 4 1; 3 8 7; 4 4 2] $S_i$ = [50 40 60] $D_j$ = [20 95 35]	0.045671	0.0430081
27	5 × 5	[24] $C_{ij}$ = [73 40 9 79 20; 62 93 96 8 13; 96 65 80 50 65; 57 58 29 12 87; 56 23 87 18 12] $S_i$ = [8 7 9 3 5] $D_j$ = [6 8 10 4 4]	0.050447	0.048615
28	4 × 6	[26] $C_{ij}$ = [25 30 20 40 45 37; 30 25 20 30 40 20; 40 20 40 35 45 22; 25 24 50 27 30 25] $S_i$ = [37 22 32 14] $D_j$ = [15 20 15 25 20 10]	0.052754	0.048678
29	3 × 4	[10] $C_{ij}$ = [8 6 10 9; 9 12 13 7; 14 9 16 5] $S_i$ = [35 50 40] $D_j$ = [45 20 30 30]	0.052999	0.047592
30	2 × 3	[10] $C_{ij}$ = [7 8 10; 9 7 8] $S_i$ = [50 50] $D_j$ = [40 40 40]	0.054106	0.050773
31	$5\times 10$	Self. $C_{ij}$ = [6 9 5 7 8 6 4 9 3 7; 7 5 8 6 9 7 5 8 6 4; 8 6 9 7 5 8 6 9 7 5; 5 8 6 9 7 5 8 6 9 7; 9 7 5 8 6 9 7 5 8 6]. $S_i$ = [35 50 45 40 30]. $D_j$ = [15 20 25 10 30 18 12 22 20 28].	0.079673	0.064325

and

Table 14. Comparison of the Computational Speed of the New Algorithm with VAM (Examples 32–34).

Example	Problem Size	Sources and Problems	VAM (Time in Seconds)	New Algorithm (Time in Seconds)
32	$10\times 10$	Self. $C_{ij}$ = [4 6 9 7 8 5 7 6 8 9; 7 4 6 9 7 8 5 7 6 8; 8 7 4 6 9 7 8 5 7 6; 6 8 7 4 6 9 7 8 5 7; 5 6 8 7 4 6 9 7 8 5; 9 5 6 8 7 4 6 9 7 8; 7 9 5 6 8 7 4 6 9 7; 6 7 9 5 6 8 7 4 6 9; 8 6 7 9 5 6 8 7 4 6; 6 8 6 7 9 5 6 8 7 4]. $S_i$ = [30 25 40 35 20 50 25 30 20 25]. $D_j$ = [35 20 25 30 15 40 25 35 30 45].	0.042330	0.037643
33	$10\times 15$	Self. $C_{ij}$ = [4 7 5 8 6 4 7 5 8 6 4 7 5 8 6; 6 4 7 5 8 6 4 7 5 8 6 4 7 5 8; 8 6 4 7 5 8 6 4 7 5 8 6 4 7 5; 5 8 6 4 7 5 8 6 4 7 5 8 6 4 7; 7 5 8 6 4 7 5 8 6 4 7 5 8 6 4; 9 7 5 8 6 9 7 5 8 6 9 7 5 8 6; 6 9 7 5 8 6 9 7 5 8 6 9 7 5 8; 8 6 9 7 5 8 6 9 7 5 8 6 9 7 5; 5 8 6 9 7 5 8 6 9 7 5 8 6 9 7; 7 5 8 6 9 7 5 8 6 9 7 5 8 6 9]. $S_i$ = [28 42 33 47 22 36 28 50 44 50]. $D_j$ = [22 27 32 36 18 42 30 24 28 20 26 38 34 28 15].	0.034432	0.032304
34	$10\times 20$	Self. $C_{ij}$ = [6 9 4 7 5 8 6 5 7 9 4 8 6 7 5 9 6 4 8 5; 7 6 8 5 9 4 7 6 8 5 9 6 5 8 7 4 6 9 5 7; 5 8 6 9 4 7 5 8 6 7 5 9 4 6 8 5 7 4 6 9; 8 5 7 6 9 5 8 4 6 9 7 5 8 6 4 7 5 8 6 7; 9 4 6 8 5 7 3 6 9 5 8 4 7 6 5 8 5 8 6 7; 4 7 5 8 6 9 7 5 8 6 9 7 4 6 9 5 8 7 5 6; 6 5 9 4 7 6 8 5 7 4 6 8 7 5 9 6 4 7 5 8; 5 7 4 6 9 7 5 7 4 6 9 7 5 7 4 6 9 7 5 7; 7 4 6 9 5 8 6 9 5 7 6 4 8 5 7 6 9 5 8 6; 6 8 5 7 4 6 9 7 5 7 4 6 9 7 5 7 4 6 9 7]. $S_i$ = [48 37 55 62 41 59 46 68 44 50]. $D_j$ = [22 22 24 28 26 20 34 25 21 27 23 29 35 23 24 26 28 24 24 25].	0.064502	0.057820

, give the results of that comparison.

5. Theoretical Justification of the Fractional Penalty

This section provides analytical clarification regarding the structural properties of the fractional penalty and its relationship to classical VAM.

5.1. Scale Invariance Property

Unlike the additive penalty used in VAM, the proposed ratio-based penalty possesses a scale-invariance property, which is formally stated below.

Proposition 1. 

Let all transportation costs be multiplied by a positive scalar $\alpha >0$. The fractional penalty

$$p_i={\displaystyle \frac{m_i^{\left(2\right)}}{m_i^{\left(1\right)}}}$$

(44)

remains unchanged under this transformation.

Proof.  

Let the transformed costs be

$${\tilde{c}}_{ij}=\alpha c_{ij}.$$

(45)

Then the smallest and second-smallest costs in any active row become

$${\tilde{m}}_i^{\left(1\right)}=\alpha m_i^{\left(1\right)},{\tilde{m}}_i^{\left(2\right)}=\alpha m_i^{\left(2\right)}.$$

(46)

Hence, the transformed penalty becomes

$${\tilde{p}}_i={\displaystyle \frac{{\tilde{m}}_i^{\left(2\right)}}{{\tilde{m}}_i^{\left(1\right)}}}={\displaystyle \frac{\alpha m_i^{\left(2\right)}}{\alpha m_i^{\left(1\right)}}}={\displaystyle \frac{m_i^{\left(2\right)}}{m_i^{\left(1\right)}}}=p_i.$$

(47)

Therefore, the ratio-based penalty is invariant under positive cost scaling.    □

This property ensures prioritization stability across heterogeneous cost magnitudes, a characteristic not shared by additive difference penalties.

5.2. Structural Relationship to VAM

The proposed method preserves the allocation structure of VAM. The only modification lies in the prioritization criterion.

Proposition 2. 

If the ranking of rows and columns induced by the ratio-based penalty coincides with the ranking induced by the additive penalty, then the proposed method generates the same IBFS as VAM.

Proof. 

If both penalties produce identical ordering of lines at every iteration, then the selected allocation sequence is identical. Since allocation feasibility rules remain unchanged, the final IBFS coincides with that of VAM.    □

This explains why identical solutions are sometimes observed in benchmark results. Improvements occur precisely when multiplicative contrast alters prioritization order.

5.3. Degeneracy Handling

As with classical VAM, degeneracy occurs when the number of allocations is fewer than $m+n-1$. In such cases, an $\epsilon$-allocation is inserted in a zero-cost admissible cell to maintain the required number of basic variables without disturbing feasibility.

Across the 34 benchmark instances, degeneracy occurred in only a limited subset of cases, and handling procedures were identical for VAM and the proposed method.

5.4. Dominance Considerations: Additive vs. Multiplicative Ordering

It is natural to ask under what conditions the multiplicative penalty produces a different prioritization order from the classical additive penalty used in VAM.

Let two candidate rows i and k have smallest and second-smallest admissible costs

$$(m_i^{\left(1\right)},m_i^{\left(2\right)})\mathrm{and}(m_k^{\left(1\right)},m_k^{\left(2\right)}).$$

(48)

The additive penalties are

$${\Delta }_i=m_i^{\left(2\right)}-m_i^{\left(1\right)},{\Delta }_k=m_k^{\left(2\right)}-m_k^{\left(1\right)}.$$

(49)

while the multiplicative penalties are;

$$p_i={\displaystyle \frac{m_i^{\left(2\right)}}{m_i^{\left(1\right)}}},p_k={\displaystyle \frac{m_k^{\left(2\right)}}{m_k^{\left(1\right)}}}.$$

(50)

If the relative dispersion between the two smallest costs is proportional across rows (i.e., if $m_i^{\left(2\right)}=\lambda m_i^{\left(1\right)}$ and $m_k^{\left(2\right)}=\lambda m_k^{\left(1\right)}$ for the same $\lambda$), then both additive and multiplicative penalties induce identical rankings.

$$m_i^{\left(2\right)}=\lambda m_i^{\left(1\right)},m_k^{\left(2\right)}=\lambda m_k^{\left(1\right)}.$$

(51)

However, when proportional contrasts differ across rows—particularly in heterogeneous cost environments—the ratio-based penalty may alter prioritization even when additive differences are similar. In such cases, multiplicative ordering emphasizes relative dominance rather than absolute deviation.

Therefore, the new method does not universally dominate VAM; rather, it modifies prioritization when proportional dispersion differs from additive dispersion. This explains why identical IBFS outcomes are sometimes observed, while improvements arise in other benchmark categories.

5.5. Discussions

The comparative analysis across thirty-four datasets shows that the fractional-penalty algorithm frequently matches or improves upon the IBFS cost obtained by NWCM, LCM, and VAM. Improvements are particularly noticeable in instances where cost structures exhibit proportional clustering or multiple near-equal minima.

The computational experiments indicate that the fractional-penalty prioritization can improve the quality of initial basic feasible solutions in several heterogeneous cost environments. Although the structural framework of the method remains consistent with classical VAM, the multiplicative penalty emphasizes proportional dispersion between competing costs rather than absolute cost differences. This distinction becomes particularly relevant when transportation costs vary significantly across rows or columns. In such cases, the ratio-based prioritization may alter allocation ordering and lead to improved initial solutions. At the same time, the proposed method retains the simplicity, transparency, and computational efficiency of classical IBFS procedures, making it suitable for practical implementation.

However, the magnitude of improvement varies across datasets. In several benchmark cases, the proposed method yields identical results to VAM, indicating structural equivalence in those instances. Runtime differences are modest and should be interpreted cautiously due to microsecond-level variability inherent in MATLAB timing measurements.

Overall, the results suggest that the fractional penalty provides an alternative prioritization mechanism that can enhance IBFS quality without increasing asymptotic complexity.

5.6. Sensitivity and Boundary Conditions

The fractional penalty behaves similarly to VAM when cost dispersion within rows/columns is nearly uniform. However, when proportional contrast between the smallest two costs is high, the ratio-based penalty can alter prioritization order.

If $m^{\left(1\right)}$ approaches zero, special care is required to avoid numerical instability; in implementation, zero-cost safeguards were included.

These observations clarify that the proposed method is best interpreted as a structured refinement whose effectiveness depends on cost dispersion characteristics.

6. Conclusions

This study introduced a ratio-based fractional penalty refinement of Vogel’s Approximation Method for constructing initial basic feasible solutions to transportation problems. By redefining the penalty as a proportional contrast measure rather than an additive difference, the method preserves the constructive allocation structure of VAM while offering scale-invariant prioritization.

Extensive benchmark testing across thirty-four datasets demonstrates that the proposed algorithm consistently produces IBFS costs that are equal to or lower than classical heuristics in many cases. Statistical evaluation confirms that improvements are systematic rather than incidental.

The method retains the same computational complexity as VAM and incorporates deterministic tie-breaking and degeneracy safeguards to ensure reproducibility. Future research may investigate hybrid additive–multiplicative penalty combinations, theoretical dominance conditions, and large-scale industrial datasets. The new method should therefore be interpreted as a structured penalty refinement within the VAM framework, supported by analytical properties and empirical validation, rather than as a paradigm replacement.

Additional computational details, extended results, and MATLAB R2022b implementations are provided in Appendix A, Appendix B, Appendix C and Appendix D.