The study evaluates four distinct modes of transportation: 1) a unimodal transport chain involving trailer transport from the origin to destination, and two multimodal transport chains utilizing; 2) rail/truck and 3) air/truck and an intermodal 4) container on flat car (COFC)/Truck for the transport. Therefore, the four potential carrier alternates are considered: A₁ (All along trailer), A₂ (Rail/Truck combination), A₃ (Air/Truck combination), and A₄ (Container on Flat Car (COFC) intermodal and truck). The configurations of alternative multimodal freight transportation are visually depicted in Figure 3 .

Definition: An intermodal becomes a multimodal when it operates under a single freight transportation contract.

Modal transportation offers a powerful tool for businesses to optimize their supply chains. By combining different modes of transport (truck, rail, ship), multimodal systems leverage the strengths of each to achieve greater efficiency and cost-effectiveness compared to traditional single-mode options [32] . As illustrated in Figure 3 , most modal systems involve three key stages:

Pre-Haulage (First Mile): Trucks are typically the most efficient way to move goods short distances from factories or warehouses to the initial transport hub (port, rail yard). Businesses often use their own trucks for this leg due to the proximity of facilities to these hubs.

Long-Haul (The Main Event): The long-distance journey utilizes the most suitable mode (rail, ship, air) or even a combination of modes depending on cost, speed, and distance. Trailers on flatcars (TOFC) and containers on flatcars (COFC) are commonly used for efficient rail transport.

Post-Haulage (Last Mile): Like pre-haulage, trucks are often the best choice for final delivery to urban centers, warehouses, and distribution centers. This “last mile” delivery is often referred to as drayage.

The distance between the main modal starts and ends is Z miles. Here, pre-haulage distance, a is considered from the origin (facility) to the main modal start point. The post-haulage distance b is from the main modal start point to the destination. Therefore, total transport travel distance = Z + (a + b).

Intermodal transport offers a strategic advantage when the total costs (including pre and post-haulage) become competitive with traditional trucking for the entire distance. Figure 4 presents pre-haulage, long haulage, and post-haulage transport cost functions.

Total Cost:

$T{C}_m=\left(\frac{a_m}{S_m}\left[U_m\right]+F_1\right)+\left(\frac{Z_m}{S_m}\left[U_m\right]+F_m\right)+\left(\frac{b_m}{S_m}\left[U_m\right]+F_2\right)+w_m\times U\left(k\right)$

Here, subscript m is modal: truck, Rail/Truck, Air/Truck, COFC/Truck etc.

where

$T{C}_m$: Total cost for modal transportation

$\frac{Z_m}{S_m}\left[U_m\right]+F_m$: Main modal transport cost ($)

$\frac{a_m}{S_i}\left[U_m\right]+F_1$: Pre-haulage transport cost ($)

$\frac{b_m}{S_m}\left[U_m\right]+F_2$: Post-haulage transport cost ($)

$W_{R/T}$: Transshipment cost

Z: Distance, (mile)

a_m: Pre-haulage distances (road transport) (miles)

b_m: Post-haulage distances (road transport) (miles)

S_m: Transport speed (transport mode) (mile/hour)

F_m: Main transport tariff, ($/Container mile)

k: Transshipment time at the intermodal terminal (hours)

U: Value of time (by transport mode) ($/Container/hour)

U(k): Transship time: Time to transfer container at modal terminal

W: Transship cost: Cost for modal transport ($/Container)

F₁: Pre-haulage tariff ($/Container mile)

F₂: Post-haulage tariff ($/Container mile)

Freight transportation modeling requires fixed and variable expenses related to transport modes, vehicle types, product groups, origin-destination data, route networks, and terminal information for transferring containers between transport modes. In Figure 3 , A₁ is unimodal freight transport by trailer; A₂ is (Rail/Truck) and there is no pre-haulage, so the distance is travelled by rail transport as the main mode and truck for the post-haulage transport. The corresponding equation for the main mode of rail freight and truck for the post-haulage transport is below.

$T{C}_{R/T}=Z_R\times \left(\frac{{\beta }_R}{h_R}+{\tau }_R\right)+b_T\times \left(\frac{{\beta }_T}{h_T}+g_2\right)+w_m+\beta \left(r\right)$

The TOPSIS method with entropy weighting helps select the best carrier among discrete performance data by objectively assigning weights to criteria (e.g., cost, time) and ranking alternatives based on their closeness to an ideal scenario. The entropy method is useful when policymakers conflict with the values of weights [19] . This approach reduces bias from subjective weighting and helps businesses find the most efficient carrier for cost savings and market advantage. The procedure involves ranking alternative carriers structured into a series of systematic steps.

Step 1: Identify Alternatives and Evaluation Criteria

Determine potential transport routes and measurable characteristics, forming a decision matrix of discrete choices between alternatives and criteria. A decision maker identify n selection criteria $C=\left(C_j|j=1,2,\cdots ,n\right)$ through which m alternative transportation modes $A=\left(A_i|i=1,2,\cdots ,m\right)$. The data is expressed in a (m × n) matrix, representing the discrete choices between the criteria and alternatives. The Freight selection MCDM matrix is shown in Table 2 :

Matrix element, ${\tilde{x}}_{ij}$ is the performance rating of criteria C_j for each alternative mode A_i. The normalized value $P_{ij}$ is calculated as

$P_{ij}=x_{ij}/\sqrt{{\displaystyle {\sum }_{i=1}^m{x}_{ij}^2}}$, $i\in \left(1,2,\cdots ,m\right)$, $j\in \left(1,2,\cdots ,n\right)$ (1)

where $x_{ij}$ represents the numerical evaluation of alternative A_i for criterion C_j. The squared normalization of the data eliminates the differences of measurement units and inconsistent scale.

Step 2: Entropy Coefficient Method to Evaluate Weight Criteria

Apply the entropy method to determine the weight $W=\left({\tilde{w}}_j|j=1,2,\cdots ,n\right)$ for each criterion, $C_j\left(j=1,2,\cdots ,n\right)$. Using the normalized decision matrix, $q_{ij}$, the entropy weight coefficient E_j is calculated as follows:

$E_j=-k{\displaystyle {\sum }_{j=1}^n{q}_{ij}\ln \left(q_{ij}\right)}$ (2)

where $q_{ij}=x_{ij}/{\displaystyle {\sum }_{i=1}^m{x}_{ij}}$, and k (constant) = 1/ln(m).

The larger the entropy within the criteria the more likely it is to be an important criterion. Note, $q_{ij}$ and $P_{ij}$ are not same. The measurement of dispersion D_j for a criterion is calculated as, the following:

$D_j=1-E_j$. (3)

The higher D_j value indicates the importance of the criterion in the decision matrix. The weight w_j for each attribute C_j is calculated by using the following formula:

$w_j=\frac{D_j}{{{\displaystyle \sum }}_{k=1}^n{D}_k}$ (4)

$w_j={\tilde{w}}_1,{\tilde{w}}_2,\cdots ,{\tilde{w}}_n$, where ${\tilde{w}}_j$ is the weight of jth criterion C_j.

Step 3: Construct Weighted Normalized Decision Matrix

Update the matrix elements by multiplying the entropy-derived weights with normalized performance ratings as follows:

$\widetilde{V_{ij}}=\widetilde{P_{ij}}\times \widetilde{w_j}$ (5)

where $\widetilde{w_j}$ is weight. $\widetilde{V_{ij}}$, $i\in \left(1,2,\cdots ,m\right)$, $j\in \left(1,2,\cdots ,n\right)$ element in decision matrix.

Step 4: Determine Intuitionistic Positive and Negative Ideal Solution.

This step is to identify the positive ideal reference point V⁺ and the negative ideal reference point V⁻. Among the evaluation criteria, J_b ( $J\in J_b$) is the set of benefit criteria, and J_c ( $J\in J_c$) is the set of cost criteria. The V⁺ and V⁻ are obtained as follows.

$V^{+}=\left\{\left({\max }_i{V}_{ij}|J\in J_b\right),\left({\min }_i{V}_{ij}|J\in J_c\right)\right\}=\left\{V_1^{+},V_2^{+},\cdots ,V_n^{+}\right\}$ (6a)

$V^{-}=\left\{\left({\min }_i{V}_{ij}|J\in J_c\right),\left({\max }_i{V}_{ij}|J\in J_c\right)\right\}=\left\{V_1^{-},V_2^{-},\cdots ,V_n^{-}\right\}$ (6b)

Step 5: Determine the Separation Measures between the Alternatives

The distance of each alternative from the positive ideal reference point ( $S_i^{+}$ from V⁺) and negative ideal reference point, ( $S_i^{-}$ from V⁻) is obtained as follows:

Positive Ideal Separation: $S_i^{+}=\sqrt{{\displaystyle {\sum }_{j=1}^{n}d}{\left(V_{ij}-V_j^{+}\right)}^2},i=1,2,\cdots ,m$ (7a)

Negative Ideal Separation: $S_i^{-}=\sqrt{{\displaystyle {\sum }_{j=1}^{n}d}{\left(V_{ij}-V_j^{-}\right)}^2},i=1,2,\cdots ,m$ (7b)

Here, $d\left(V_{ij}-V_j^{+}\right)$ represents the distance between a given element and the maximum value. $S_i^{+}$ denotes the distance of an alternative from V⁺ (the positive ideal solution), while $S_i^{-}$ indicates the distance of the alternative from V⁻ (the negative ideal solution).

Step 6: Obtain the Closeness Co-efficient to Rank the Alternatives.

The TOPSIS method assigns a closeness index (U_i) to each carrier, indicating its proximity to the ideal option. The value of U_i is obtained as follows:

$U_i=\frac{S_i^{-}}{S_i^{+}+S_i^{-}},i=1,2,\cdots ,m;0\le U_i\le 1$. (8)

A U_i closer to 1 signifies a higher priority carrier, as it’s closer to the best possible scenario and furthest from the worst.

The illustrative case is a data-driven approach that offers the selecting most suitable carrier for the specific freight scenario. The numerical findings of the case study illustrate the practical application of the TOPSIS method integrated with the entropy weight coefficient technique, facilitating the thorough analysis of carriers’ selection from a range of multimodal/intermodal transportation alternatives. The total cost of transportation for each modal includes pre-haulage, main modal, and post-haulage transport costs, along with transshipment costs. The distance between the point of origin and destination, equivalent to California and Midwest city (Chicago), is assumed to be 2000 miles. Table 3 provides the comparative intermodal cost elements.

The distance traveled by a primary transport mode is Z + (a + b), where Z is the long haulage distance and a and b are the pre-haulage and post-haulage distance, respectively. Road transport only uses pre- and post-haulage. Freight transportation costs fluctuate over time due to fuel prices, infrastructure maintenance, and market conditions. The context for cost evaluation includes the average freight transportation prices (percent/ton-mile) across different modes are $2.50 for the railway, $25.08 for truck/motorway, $0.73 for water/seaway, and $58.75 for airway. Total transport cost is represented by the cost derived using Equation (1). The transport cost comparison of the selected freight transport mode is shown in Table 4 .

Following Step 1, the performance rate of each carrier alternative and the corresponding criteria are presented in Table 5 . The freight carrier alternatives A_j(j = 1, 2, ···, 4), evaluated against the selected attributes C_i(i = 1, 2, ···, 4), the data is expressed in a (4 × 4) matrix that represents the discrete choice between the attributes and alternatives.

Squired normalization value $P_{ij}=x_{ij}/\sqrt{{\displaystyle {\sum }_{i=1}^m{x}_{ij}^2}}$., presented in Table 6 .

COFC* = Container on Flat Car, (Rail-Truck).

For weight calculation using the entropy method, we used normalized method such that $p_{ij}=x_{ij}/{\displaystyle {\sum }_{i=1}^m{x}_{ij}}$. Here, the number of criteria, m = 4. Following is the entropy weight coefficient discussed in Step 2, presented in Table 7 .

In Table 7 , Row 1, the value E_j is calculated using Equation (2),

$E_j=-k{\displaystyle {\sum }_{j=1}^n{q}_{ij}\ln \left(q_{ij}\right)}$. Therefore, the value of $P_i$ for $T_{j=1}$ with respect to

attributes A₁, A₂, A₃ and A₄ are 5/25, 7/25, 5/25, and 8/25, respectively.

$E_{j=1}$ is calculated as follows: $-\left(0.20\times \ln 0.20+\cdots +0.32\times \ln 0.32\right)/\ln 4=$ $0.9863$. High entropy suggests that criteria with considerable variation across options play a more crucial role in decision-making.

In Table 7 , Row 2, following Equation (3), $D_j=1-E_j=1-0.9863=0.0155$. In Table 7 , Row 3, the weight, W_j using Equation (4). ${\displaystyle {\sum }_{j=1}^n{D}_j}=0.0155+0.0155+0.00350+0.0137=0.0757$. Thus, $W_j=0.0155/0.0757=0.2046$.

To construct the weighted normalized decision matrix, as described in Step 3, the weight factors (determined in Step 2 and listed in the last row of Table 7 multiplied with the normalized value $P_{ij}$ (obtained in Step 1 and listed in Table 6 ) to produce the results shown in Table 8 .

In Table 8 , we calculate the following equation: $\widetilde{V_{ij}}=\widetilde{Q_{ij}}\times \widetilde{w_j}$. Using values in Table 6 and Table 7 , the cell, $V_{11}=A_1\times C_{w1}=0.3666\times 0.2046=0.0821$, given A₁ (Trailer) and C₁ (Cost). The reference points discussed in Step 4 are the Positive Ideal Reference Point (v⁺) and Negative Ideal Reference Point (v⁻). From Table 8 , the v⁺ and v⁻ are the maximum and minimum values corresponding to each criterion, shown in Table 9 .

Criteria 2-3 are benefit criteria meaning the highest value is preferred. Criteria 1 is a cost criterion meaning the lowest value is preferred. Calculation in Step 5 represents the distance from the ideal reference points. The S_i⁺ and S_i^- follows the Equation (7a) and Equation 7(b).

$S_i^{+}=\sqrt{\underset{j=1}{\overset{n}{{\displaystyle \sum }}}{\left(v_{ij}-v_j^{+}\right)}^2}$ ( $v_{ij}$ from Table 6 , and $v_j^{+}$ from Table 7 )

$S_i^{-}=\sqrt{\underset{j=1}{\overset{n}{{\displaystyle \sum }}}{\left(v_{ij}-v_j^{-}\right)}^2}$ ( $v_{ij}$ from Table 6 , and $v_j^{-}$ from Table 7 )

The relative closeness of the alternate transport modes concerning S⁺ and S⁻ has been defined in Equation (8) as $U_i=\frac{S_i^{-}}{S_i^{+}+S_i^{-}}$ for $i=1,2,\cdots ,m$; $0\le U\le 1$. Since S⁺ > 0 and S⁻ > 0, the relative closeness index, $C_i\in \left[0,1\right]$. The $S_i^{+}$ and $S_i^{-}$ and closeness index U_i, in Step 6 for each alternate carrier mode ranking, is presented in Table 10 .

The selection of freight involves optimizing delivery time, cost, and the ability to transport products effectively. Factors like cost, timeliness, and damage prevention are key when choosing freight. As shown in Table 8 , the Rail/Truck combo mode ranks number one, being the most economical, widely used, reliable, and flexible option with on-time service. In this numerical study, the COCF (Container on Container Freight) mode ranks second best, presenting a viable alternative to the Rail/Truck mode. As companies grow within their supply

chain industries, they increasingly recognize the benefits of managing transportation processes through strategic contract negotiations and optimizing costs, reliability, time sensitivity, and carrier capabilities.