One of the first uses of linear programming is the transportation problem (TP). It handles the movement of products between many supply points (sources) and several demand locations (destinations). It is possible to construct a wide range of decision issues as TPs, including production distribution, job scheduling, inventory control, and investment analysis. Determining the delivery plan that minimizes overall shipping costs while meeting supply and demand restrictions is the primary goal of the transportation challenge. In operations management, the application of transportation problems for variations in cost is crucial for optimizing supply chain logistics and distribution networks. Let’s consider a manufacturing company that sources raw materials from multiple suppliers and distributes finished products to diverse markets. Fluctuations in fuel prices, labor costs, or other variables can significantly impact transportation expenses. By employing transportation optimization models, operations managers can design efficient delivery routes, select appropriate carriers, and allocate resources effectively. In the face of cost variations, the system can dynamically adjust to minimize transportation expenses. For instance, if fuel costs rise unexpectedly, the optimization model might recommend consolidating shipments, re-routing deliveries, or even exploring alternative transportation modes to maintain cost efficiency. This application of transportation problems in operations management ensures that the supply chain remains agile and responsive to changing cost dynamics. It not only helps control operational expenses but also contributes to overall efficiency and customer satisfaction by ensuring timely and cost-effective deliveries. This approach is particularly relevant in industries where transportation costs form a significant portion of the overall operational expenses, such as manufacturing, retail, and distribution. The foundational transportation model, initially proposed by Hitchcock [1] , underwent advancements by Koppmans [2] and Danzig [3] . Several heuristic techniques, including the North-West corner rule, least cost approach, Vogel’s approximation method (VAM) [4] , Russell’s approximation method [5] , and other renowned heuristics, were employed to derive a simple and practical solution. Kirca and Satir [6] introduced the total opportunity cost approach (TOM) as a heuristic for obtaining the initial viable solution, later expanded by Mathirajan and Meenakshi [7] using VAM. Korukoglu and Balli [8] proposed an enhanced VAM based on total opportunity cost (TOC). Charnes and Cooper [9] devised the stepping stone method, while Dantzig [10] introduced the modified distribution (MODI) approach for verifying optimality in the initial fundamental solution. Numerous scholars, including Amaliah et al. [11] , Hosseini [12] , Jude et al. [13] , Juman and Hoque [14] , Karagul and Sahin [15] , and Uddin and Khan [16] , have delved into the exploration of Initial Basic Feasible Solution (IBFS) as a means to directly attain the optimal solution for Transportation Problems (TP) without relying on Stepping Stone or Modified Distribution (MODI) methods. Rashid [17] contributed a theorem in the context of resolving transportation issues. Ahmed et al. [18] introduced an incessant allocation method to analyze and minimize transportation cost. Taha [19] , studied for finding optimal solution of transportation problems. Amaliah et al. [20] , introduced a heuristic method to find the initial basic feasible solution of transportation problem (TP).

In the existing literature, the authors obtain that the optimal solution of Transportation Problems (TP) without incorporating changes to the Transportation cost. In general, various communication systems in real-world applications demand representation through variations in Transportation cost, particularly for multiservice facilities catering to customers. This investigation addressed in this paper by integrating the variation of Transportation cost. The presented models emerge as a more reliable framework when compared to the existing models in the literature. An efficient transportation plan not only reduces costs but also improves overall operational efficiency. This can lead to reduced lead times, faster order fulfillment, and enhanced customer satisfaction. The main objective of this paper is to examine the impact of fluctuations in transportation costs on reliability. Significant changes in costs can disrupt the optimal transportation plan and impact various aspects of an organization’s operations. Adapting to these changes effectively through cost management strategies, contingency planning, and informed decision-making is crucial for maintaining efficiency, competitiveness, and financial stability in dynamic business environments. These two suggested theorems have the following benefits:

1) Analyzing the effect on transportation expenses of possible disruptions or modifications in the availability of resources.

2) The two suggested theorems in the business sector can be used to examine how sensitive transportation costs are to modifications in the component parts of the cost matrix.

3) Quickly modifying transportation expenses without sacrificing the best possible outcome in order to effectively manage resources.

4) Compared to starting from scratch and solving the problem, finding the best solution for the modified problem typically requires less computing work.

This paper is organized as follows: Section 2: mathematical formulations; Section 3: description of the models; Section 4: analysis of the proposed theorems; Section 5: numerical examples; Section 6: discussion of results; and the last section: the conclusion of the paper.

Let m suppliers (sources) and n customers (destinations) be considered. The m suppliers can ship a single homogeneous product to any of the n customers at a shipping cost per unit c i j . Let a i represent the number of supply units needed at source i ( i = 1 , 2 , ⋯ , m ) , b j represent the number of demand units needed at destination j ( j = 1 , 2 , ⋯ , n ) and x i j represent the unknown quantity that needs to be transported from source i to destination j. Consequently, the corresponding linear programming model will be

Minimize (total cost) z = ∑ i = 1 m ∑ j = 1 n c i j x i j

Subject to

∑ j = 1 n x i j = a i     for   i = 1 , 2 , ⋯ , m     ( supply   constraint ) ∑ i = 1 m x i j = b j     for   j = 1 , 2 , ⋯ , n     ( demand   constraint ) x i j ≥ 0.

In matrix form the model can be formulated as Table 1.

In this section, we have shown two modified cost matrix using the addition or subtraction of constants α i ; i = 1 , 2 , ⋯ , m and β j ; j = 1 , 2 , ⋯ , n to specific rows or columns within the matrix, along with the multiplication of constants k within the same cost matrix. Through these systematic modifications, the impact of changes to individual elements on the overall transportation cost structure can be precisely measured (Table 2 and Table 3).

The total transportation cost of modified cost matrix for TP is minimized by the following formula for Table 2.

z * = ∑ i = 1 m ∑ j = 1 n ( c i j ± α i ± β j ) x i j

Therefore, the total cost (objective function) becomes for Table 3.

z * = k ∑ i = 1 m ∑ j = 1 n c i j x i j

Different models have different assumptions and approaches. Using two models provides a more robust analysis that accounts for various scenarios and conditions. It helps identify the potential impact of model-specific assumptions on the results. These models help in understanding how variations in transportation costs affect the optimal solution under different modeling frameworks. In this section we have proposed two theorems to find the effect of optimal solution for transportation problem using variation of costs.

Theorem 4.1: The optimality of the solution to the original problem is maintained for the new problem if there is an addition or subtraction of a constant quantity α i from each element of the ith row and/or a constant quantity β j from each element of the jth column of the transportation cost matrix [ c i j ] . Additionally, the total transportation cost is altered by

∑ i = 1 m ( ± α i ) a i + ∑ j = 1 n ( ± β j ) b j .

Proof: Let { x i j : i = 1 , 2 , ⋯ , m   ; j = 1 , 2 , ⋯ , n } represent the optimal solution in relation to the original cost matrix [ c i j ] . Then, the objective function for total cost is provided by

z = ∑ i = 1 m ∑ j = 1 n c i j x i j

If the real constants α i and β j are added (or subtracted) to (or from) ith row and jth column respectively of the cost matrix [ c i j ] and z * indicates the entire cost of the modified cost matrix, [ c i j ± α i ± β j ] after that

z * = ∑ i = 1 m ∑ j = 1 n ( c i j ± α i ± β j ) x i j

= ∑ i = 1 m ∑ j = 1 n c i j x i j + ∑ i = 1 m ( ± α i ) ∑ j = 1 n x i j + ∑ j = 1 n ( ± β j ) ∑ i = 1 m x i j = z + ∑ i = 1 m ( ± α i ) a i + ∑ j = 1 n ( ± β j ) b j

Since the terms that are added to z to give z * are independent of x i j , it follows that z * is minimized whenever z is minimized or vice versa and the total

transportation cost is changed by ∑ i = 1 m ( ± α i ) a i + ∑ j = 1 n ( ± β j ) b j .

Theorem 4.2: If a constant amount is multiplied by each element in a transportation cost matrix [ c i j ] , then the overall transportation cost is increased by the constant quantity times, and the best solution of the original problem remains optimal for the new problem.

Proof: Let { x i j : i = 1 , 2 , ⋯ , m   ; j = 1 , 2 , ⋯ , n } represent the optimal solution in relation to the original cost matrix [ c i j ] . Then, the objective function for total cost is provided by

z = ∑ i = 1 m ∑ j = 1 n c i j x i j

If each element of the cost matrix [ c i j ] is multiplied by a constant quantity k then the total cost (objective function) becomes

z * = ∑ i = 1 m ∑ j = 1 n ( k c i j ) x i j = k ∑ i = 1 m ∑ j = 1 n c i j x i j = k z

Hence if x i j is the optimal solution of the cost matrix [ c i j ] then it is also optimal for the cost matrix [ k c i j ] and the total cost is increased by k times.

In real-life scenarios, the transportation problem can be found in various contexts such as supply chain management, logistics, and distribution planning, where minimizing transportation costs while meeting supply and demand requirements is a critical objective. These theorems provide practical methods to address these optimization challenges. Subsequently, we provide numerical examples to validate the introduced theorems.

Example 5.1: We investigate the following transportation problem to demonstrate Theorem 1.

To find the optimal cost of the TP we have used least cost method and then employing MODI method. The optimal transportation schedule is presented as Table 4.

Consequently, the optimal solution and the aggregate transportation cost are

x 12 = 130 ,     x 14 = 20 ,     x 21 = 140 ,     x 24 = 120 ,     x 32 = 270 ,     x 33 = 250 z = 25910.

We observe the following, when we modified the cost matrix of original transportation problem:

If we add α 1 = 14 , α 2 = 2 , α 3 = 10 , β 1 = 4 , β 3 = 9 and subtract β 2 = 12 , β 4 = 14 to each unit cost in the original transportation problem by [ c i j ± α i ± β j ] , the modified cost matrix is given below:

To find the optimal cost of the modified TP we have also used least cost method and then applying MODI method. The optimal transportation schedule is as Table 5.

Consequently, the optimal solution and the aggregate transportation cost are

x 12 = 130 ,     x 14 = 20 ,     x 21 = 140 ,     x 24 = 120 ,     x 32 = 270 ,     x 33 = 250 z * = 29780.

Thus we see that after adding or subtracting constant quantities α i and β j , the optimal schedule remains unchanged and the total transportation cost is changed by

d = ∑ i = 1 3 ( ± α i ) a i + ∑ j = 1 4 ( ± β j ) b j         = α 1 a 1 + α 2 a 2 + α 3 a 3 + β 1 b 1 − β 2 b 2 + β 3 b 3 − β 4 b 4         = 14 × 150 + 2 × 260 + 10 × 520 + 4 × 140 − 12 × 400 + 9 × 250 − 14 × 140         = 3870 i . e .   ,     d = z * − z = 29780 − 25910 = 3870

Table 6 shows some random variations in costs and impact on optimal solution and total cost. The results are investigated through TORA optimization software.

Example 5.2: We investigate the following transportation problem to demonstrate Theorem 2.

Applying Vogel’s approximation method for initial solution and then implementing MODI method the optimal transportation schedule is as Table 7.

Consequently, the optimal solution and the aggregate transportation cost are

x 11 = 30 ,     x 12 = 170 ,     x 13 = 50 ,     x 21 = 100 ,     x 33 = 160 z = 3880.

If we multiply each element of [ c i j ] by a constant quantity k = 5 the modified cost matrix [ k c i j ] is

Solving the above problem by using Vogel’s approximation method for initial solution and then employing the MODI method, the optimal transportation table is shown in Table 8.

Consequently, the optimal solution and the aggregate transportation cost are

x 11 = 30 ,     x 12 = 170 ,     x 13 = 50 ,     x 21 = 100 ,     x 33 = 160 z * = 19400.

Thus it is observed that if we multiply each element of [ c i j ] by 5, the optimal schedule remains unchanged and the total transportation cost is increased by 5 times i.e. z * = k z = 5 × 3880 = 19400 .

Following Table 9 shows some random variations in costs and impact on

optimal solution and total cost. The results are investigated through TORA optimization software.

In the dynamic landscape of operations management, decision-makers face the challenge of navigating uncertainties, especially in the realm of transportation costs. The primary objective is to enhance decision-making by conducting comprehensive sensitivity analyses. We observed from Table 6 that when we changed the values of parameters α 1 , α 2 , α 3 , β 1 , β 2 , β 3 and β 4 the optimal solution, x 12 = 130 , x 14 = 20 , x 21 = 140 , x 24 = 120 , x 32 = 270 and x 33 = 250 , remained unchanged, but the total transportation cost changed by a significant amount d = z * − z . And also, we observed that from Table 9, when we changed the values of parameters k, the optimal solution x 11 = 30 , x 12 = 170 , x 13 = 50 , x 21 = 100 and x 33 = 160 remain unchanged, but total transportation cost is increased by k times i.e., z * = k z . Significant changes in costs can have a profound impact on the optimal transportation plan. When costs decrease significantly, it may open up opportunities for cost savings and efficiency improvements. For example, if transportation costs increase, it may be necessary to allocate more resources or budget to transportation to maintain service levels. Conversely, if costs decrease, resources may be reallocated to other areas of the business. Transportation costs are closely tied to service quality and customer satisfaction. Significant cost changes can affect delivery times, shipping options, and overall service quality. Analyzing cost changes can help organizations align their transportation strategies with sustainability goals. The sensitivity analysis is vital tools in the real world for understanding and managing the impact of cost variations on transportation problems. These procedures enable organizations to make informed decisions, optimize resource allocation, and remain adaptable in a dynamic business environment.

In this paper, we have provided two approaches (theorems) for solving transportation problems and proved them in a simple and easy way. This method provides a detailed understanding of how adjustments to specific parameters influence the financial aspects of transportation, offering valuable insights for strategic decision-making in transportation management. To present a clear overview of the theorems introduced herein, two illustrative numerical examples have been selected to demonstrate the verification of the theorems. Different models may emphasize different aspects of the Transportation Problem. By using two models, analysts can gain a more comprehensive understanding of how variations in transportation costs impact the problem from different angles, considering numerical factors. The model can be developed for further study to effectively deal with unbalanced transportation problems.

The authors declare no conflicts of interest regarding the publication of this paper.

Rashid, A. and Islam, Md.A. (2023) Analysis of the Impact of Optimal Solutions to the Transportation Problems for Variations in Cost Using Two Reliable Approaches. American Journal of Computational Mathematics, 13, 607-618. https://doi.org/10.4236/ajcm.2023.134033