Symmetry-Based Route Optimization for International Land Logistics Using an Extended Traveling Salesman Problem with Distance–Time Constraints and Real-Time Google Maps Data

Abstract

This study develops novel mathematical models to capture the complexities of international land logistics by extending the classical Traveling Salesman Problem (TSP) within a symmetry-aware optimization framework. A focused review of literature provides the theoretical basis for model formulation and highlights the limitations of conventional distance-only approaches. In international transport, shorter routes are often assumed to reduce energy use; however, this assumption overlooks the decisive influence of travel time and traffic variability. In this context, symmetry offers a useful analytical lens, as balanced relationships among distance, time, and fuel consumption can reveal more efficient logistics structures. Accordingly, two models are proposed: the Traditional Traveling Salesman Problem in terms of Distance Concentration (TTSPD), which minimizes route length, and the Extended Traveling Salesman Problem in terms of Distance and Time Concentration (ETSPDT), which jointly considers distance, travel time, and fuel consumption. Furthermore, TTSPD was employed to validate ETSPDT, since it is based on the traditional TSP. Both models are solved exactly using the Solver Add-in in Microsoft Excel 2024 with data derived from Google Maps. The results show that ETSPDT achieves superior energy efficiency and average speed, demonstrating the practical value of multidimensional, symmetry-informed optimization for sustainable supply chain and logistics management.

1. Introduction

Contemporary competitive advantage in land-based supply chain operations is increasingly shaped by the capabilities of road freight service providers [1,2,3]. Empirical evidence suggests that manufacturing firms are progressively outsourcing transportation-related activities, functions that typically fall outside their core competencies, to third-party logistics (3PL) providers. This strategic reallocation allows firms to concentrate more intensively on their primary strengths, particularly production and service delivery, thereby strengthening their competitive position and operational focus [3,4,5]. In this regard, land transport service providers may be regarded as 3PL entities that are responsible not only for the physical movement of goods, but also for the coordination and integration of associated information flows. Their role is critical in facilitating the timely, accurate, and efficient delivery of raw materials and finished products to designated production facilities, while adhering strictly to client specifications and scheduling requirements [5,6,7].

For instance, when a land transportation provider fails to deliver goods or raw materials from suppliers to end users within the required timeframe, a manufacturer’s production capacity may be severely constrained, as inputs cannot be converted into finished goods according to schedule. Such delays interrupt the continuous movement of industrial products across downstream supply chain stages, thereby undermining operational efficiency and supply chain continuity. More broadly, land transportation providers represent a pivotal node in industrial supply chains because their performance directly shapes production outcomes and logistics reliability. In this context, symmetry may be interpreted as balanced and coordinated interdependence among supply chain actors, such that disruptions at one point produce disproportionate effects throughout the system. Accordingly, symmetry offers a valuable analytical lens for examining structural alignment, temporal consistency, and coordinated functionality in supply chain resilience and performance.

A principal challenge confronting international road freight service providers lies in the strategic selection of the most efficient transportation routes, particularly those that minimize total travel distance. Because land transportation costs are strongly correlated with distance traveled, largely due to fuel consumption, route optimization constitutes a critical operational priority. In practice, international freight operations typically involve fixed delivery destinations and predetermined shipment volumes, as specified by consignors [8,9,10]. Accordingly, the central logistical objective is to design routing strategies that reduce travel distance while simultaneously ensuring timely delivery. Achieving such optimization not only strengthens a firm’s competitive position in the freight transportation sector but also fosters greater integration and coordination across manufacturing and distribution activities, thereby enhancing the overall efficiency and responsiveness of the supply chain.

International land freight operations are often organized through contractual arrangements that establish fixed shipment volumes and predetermined routes. Typically, such systems involve a single origin and a single destination, without intermediate loading or unloading points beyond the designated endpoints [11,12,13]. From a structural perspective, this configuration exhibits a form of symmetry closely related to the classical Traveling Salesman Problem (TSP), in which a vehicle departs from one location, visits a sequence of designated delivery nodes exactly once, and ultimately returns to its point of [14,15,16]. Accordingly, this inherently ordered and balanced configuration can be interpreted as a form of operational symmetry, in which the routing structure preserves a consistent relationship between origin, sequence of service, and return path. Accordingly, the use of TSP-based optimization methods in international freight logistics offers a rigorous means of minimizing transportation distance, reducing operational inefficiencies, and strengthening the competitiveness of land freight operators. At the same time, such approaches contribute to the coordinated performance of global manufacturing supply chains by improving route design, resource utilization, and overall logistical coherence.

Small-scale industrial enterprises often face intense pressure to minimize transportation costs while ensuring the timely and reliable delivery of materials to manufacturing partners [17,18]. In practice, these firms must coordinate raw material flows from multiple geographically dispersed sources, often through third-party courier networks, to downstream production facilities. This task requires strict adherence to delivery schedules and the careful management of handling constraints associated with small shipment volumes, thereby creating substantial logistical complexity. From a theoretical perspective, this setting corresponds closely to the classical transportation problem, which seeks the optimal allocation of commodities from supply nodes, such as producers and raw material suppliers, to demand nodes, with the objective of minimizing total transportation cost under supply, demand, and capacity constraints [19,20,21]. Within this framework, transportation optimization provides a rigorous analytical basis for improving both cost efficiency and operational reliability in small-scale industrial logistics systems.

More broadly, the balanced allocation of resources in such networks reflects the principle of symmetry, understood here as the harmonization of supply and demand relations across spatially distributed nodes. This symmetry-oriented perspective is especially relevant in contemporary supply chain and logistics management, where the integration of optimization, network equilibrium, and operational coordination can enhance overall system performance. Accordingly, embedding transportation optimization within raw material logistics supports efficient resource allocation and contributes to more resilient, coordinated, and conceptually coherent supply chain architectures.

Historically, international land transportation has been evaluated mainly by distance minimization, whereas temporal efficiency has often been overlooked. Yet transport networks operate in dynamic and uncertain environments, where the shortest route is not always the fastest. Congestion, flooding, accidents, and other disruptions can substantially reshape route performance. In supply chain and logistics management, therefore, spatial and temporal dimensions should be assessed together to obtain a more valid measure of efficiency. This approach reflects the principle of symmetry, which calls for the balanced consideration of interdependent factors in decision-making. Accordingly, route selection should rely on composite metrics, such as the travel-time-to-distance ratio, rather than geometry alone.

This study seeks to develop robust mathematical models to optimize international product delivery in cross-border land logistics, with particular attention to symmetry in supply chain and logistics management. It extends the classical Traveling Salesman Problem by integrating both temporal and spatial dimensions, thereby capturing the operational complexities of international road-based export activities. The proposed framework is then evaluated against the conventional distance-based TSP to assess its comparative effectiveness. The findings are expected to offer strategic guidance for logistics planners and contribute to more efficient, cost-effective, and timely delivery systems. The anticipated outcomes are expected to provide strategic insights for logistics service planners, fostering innovative approaches to streamline international supply chain operations and promote faster, more cost efficiency regarding product delivery in the context of international land transport.

2. Literature Review

The Traveling Salesman Problem (TSP) is a paradigmatic optimization challenge that seeks the shortest route through a given set of cities, each visited exactly once before returning to the origin. Originally formulated as an abstract model of route optimization, it has profoundly shaped the development of the Vehicle Routing Problem (VRP) and related logistics research. In supply chain and logistics management, the TSP underpins applications such as distribution network design, transportation planning, and delivery scheduling. Moreover, its structural symmetry, expressed through interchangeable city permutations and equivalent route reversals, has attracted substantial scholarly attention. Such symmetry not only reduces redundant search spaces but also informs the design of more efficient algorithms. Accordingly, the TSP remains central to contemporary academic inquiry and advanced operational optimization.

Recent studies have examined the computational difficulty of large-scale Travelling Salesman Problem (TSP) instances through heuristic methods. In this regard, ref. [22] employed the Greedy Algorithm to obtain near-optimal routes, showing that efficient solutions can be derived within limited computational time. Similarly, ref. [23] extended the Greedy approach to large TSP instances and integrated four RTX A4000 GPUs, demonstrating that hardware acceleration substantially improved computational speed and enabled the rapid identification of optimal or near-optimal solutions. In addition, ref. [24] proposed a K-means clustering framework to pre-group nodes before TSP optimization, thereby improving efficiency and reducing total travel distance. Collectively, these studies highlight the value of heuristic optimization in addressing complex routing problems. Moreover, when viewed through the lens of symmetry, such approaches suggest that structural regularities can be exploited to enhance solution efficiency in supply chain and logistics management, offering promising directions for advanced academic research and practical deployment.

Subsequent research has increasingly examined the Travelling Salesman Problem (TSP) through metaheuristic methods designed to address complex routing and logistics challenges. In this context, symmetry offers a valuable analytical perspective by revealing structural regularities that can simplify solution spaces and improve computational efficiency in supply chain and logistics management. For example, Genetic Algorithms (GA) have been shown to reduce travel distance and enhance logistical performance [25], while hybrid GA–Genetic Programming approaches use stochastic initialization and iterative refinement to approximate high-quality solutions more effectively [26]. Similarly, combining Ant Colony Optimization (ACO) with GA in distribution settings, such as cement delivery, underscores the strong dependence of solution quality on problem-specific conditions [27,28]. Collectively, these studies demonstrate the practical relevance of symmetry-aware optimization in advancing combinatorial decision-making.

Ref. [29] applied exact algorithms to optimize transportation time in intelligent vehicle network distribution, integrating temporal and cost variables within a Traveling Salesman Problem framework. Their results revealed substantial performance differences across sunny, snowy, rainy, and foggy conditions. Subsequently, ref. [30] expanded this line of research by proposing a Pareto-based multi-objective decision-making method, showing that total, product, and quadratic weighting schemes produce different rankings of Pareto-optimal solutions. Together, these studies highlight the importance of accounting for environmental variability and multi-criteria weighting in robust path-optimization strategies.

Complementing these findings, ref. [31] provided a comparative evaluation of exact algorithms, linear programming, heuristics, and metaheuristics on a standardized TSP dataset. Although exact methods produced globally optimal solutions, heuristic and metaheuristic techniques consistently achieved near-optimal results at much lower computational cost. Building on these advances, refs. [32,33] introduced dynamic programming and Delaunay graph simplification, respectively, demonstrating their potential to improve computational efficiency in exact and linear programming approaches. In symmetric supply chains and logistics systems, such methodical balance between optimality and efficiency remains essential.

Extensive literature indicates that exact algorithms can yield globally optimal solutions to the Traveling Salesman Problem (TSP), whereas heuristic and metaheuristic methods generally produce near-optimal approximations. As an NP-hard problem, the TSP exhibits computational complexity that grows sharply with dataset size, markedly extending solution times [32,33,34,35,36]. This challenge has driven considerable research aimed at enhancing algorithmic efficiency, reflecting ongoing efforts to balance solution optimality with practical tractability.

On the other hand, several research articles have introduced multi-dimensional models in transportation optimization. In particular, the initial five-dimensional TSP model attempts to represent data across multiple dimensions before route selection is made. For example, this optimization approach aggregates factors such as travel time, cost, risk, and other relevant criteria, and then combines these results to determine the most suitable route. However, this method has notable limitations [37,38,39]. First, it cannot fully integrate all information into a single exact representation, as ambiguity in the definition of individual dimensions and the potential for improper integration may produce distorted outcomes. Second, the resulting output often provides only an approximation, lacks guarantee of optimality, and can be difficult to interpret and visualize in real-world operational settings. Moreover, although previous studies have examined optimization across different dimensions, these dimensions are typically calculated separately before optimization, making the resulting solutions difficult to interpret and apply in practical transportation operations. Consequently, further research is needed to develop a method that can accurately integrate multiple transportation dimensions while remaining readily applicable in real transportation contexts through the use of available real-time data.

Over recent decades, route optimization research has primarily emphasized minimizing travel distance, reflecting a traditional concern with spatial efficiency. By comparison, relatively few studies have systematically examined travel time, and even fewer have jointly considered distance and time while explicitly incorporating vehicle energy consumption, a key proxy for transportation cost in logistics. This imbalance has created a notable gap in the literature, especially in relation to a more symmetric and holistic evaluation of operational efficiency. Such an integrated perspective is increasingly important in supply chain and logistics management, where the balanced treatment of temporal, spatial, and energetic factors can improve decision-making.

Addressing this gap is essential, as the interplay among travel distance, travel time, and fuel consumption directly shapes cost efficiency, environmental performance, and long-term logistical sustainability. Within the context of symmetry and supply chain and logistics management, this study proposes a comprehensive framework for the traveling salesman problem (TSP) that jointly considers these dimensions. By integrating distance, time, and vehicle fuel factors, the framework offers fresh insights into land-based logistics optimization. It enables decision-makers to enhance operational efficiency, reduce costs, and promote sustainable transportation, while also advancing conceptual understanding of symmetry-based approaches in order to strategically manage operations in the complex logistical networks.

2.1. International Land Logistics

International land logistics is a fundamental element of global supply chain management, involving the organized movement of goods across national borders through terrestrial transport systems. From a network perspective, it may be represented as a multi-node, multi-destination flow structure in which cargo moves from origin points to diverse foreign destinations. Land logistics service providers play a central role in coordinating and executing these flows by optimizing routing, scheduling, and resource allocation under uncertain conditions. In this context, symmetry offers a useful analytical lens, as balanced network structures, reciprocal flows, and equitable allocation patterns can improve efficiency, robustness, and resilience. Accordingly, symmetry-based approaches are increasingly valuable for enhancing the reliability and resilience of cross-border supply chains.

International road freight transportation involves the movement of goods for which cargo volumes are contractually fixed, with operations typically defined by a single origin and destination [11,12,13]. Conceptually, this logistical framework bears strong resemblance to the Traveling Salesman Problem (TSP). In this context, a freight vehicle departs from a domestic hub carrying a specified quantity of goods or raw materials, as determined by pre-established production requirements, and subsequently visits multiple international destinations each corresponding to a manufacturing site or organizational facility requiring a single delivery. After completing all delivery obligations, the vehicle returns to its country of origin [14,15,16]. Accordingly, the integration of TSP-based optimization techniques into international road freight operations offers significant potential to enhance the competitive positioning of transport operators, while simultaneously improving the efficiency, reliability, and responsiveness of global supply chains.

In summary, international land logistics is commonly understood as cross-border road freight transport, in which goods move from a supplier’s origin to multiple destination points across national boundaries. Such operations usually follow direct, non-return routes and are governed by formal shipping documentation and country-specific regulations. Because national transport rules, especially speed limits, vary considerably, compliance with legal velocity thresholds is essential. This means that optimization must account not only for spatial distance, but also for travel time and regulatory constraints. In this regard, the Extended Travelling Salesman Problem (ETSP) offers a rigorous framework for integrating routing decisions, time-dependent restrictions, and vehicle energy consumption into a single optimization model. Compared with the classical Travelling Salesman Problem, the ETSP better reflects the complexity of real-world logistics and supports more balanced decisions. From the perspective of symmetry, such models are particularly valuable in supply chain optimization, where structural balance can enhance efficiency, robustness, and planning consistency.

2.2. Traveling Salesman Problem: TSP

TSP is classically conceptualized as the transport optimization problem that seeks to determine the most efficient route for a salesman required to visit a given set of cities exactly once before returning to the origin. Over the years, the TSP has emerged as a fundamental framework for minimizing the total distance or associated cost of traversing all specified locations, thereby serving as a prototypical example of a minimization problem in operations research. Within this context, notations of indices and parameters in TSP can be described as in

Table 1. Notations of indices and parameters in TSP. Source: the Authors.

Notations	Description
$j,k$	Indexes present the delivery destinations $j$ or $k$ when $j, k$ = 1, 2, 3, … n
$j$	Each customer located at destination node $j$
$d_{jk}$	The transport distance from location j to location $k.$
$n$	The number of destinations in this international road freight transport trip.
$S$	The number of destinations in the designated route.
$V$	This refers to all destination.
$y_{jk}$	This refers to the transport journey from location j to $k$ when $y_{jk}$∈ {1,0}
$TSP\left(y\right)$	Total transportation distance in one trip.

, and the canonical mathematical formulation of the TSP can be formally articulated as follows.

The Traveling Salesman Problem (TSP) is conventionally formulated in terms of the pairwise transport distances associated with a given delivery itinerary. This formulation is primarily employed to optimize the allocation of products across a predefined set of terminal nodes or usage points. The TSP principal objective is to minimize the cumulative travel distance, thereby identifying the optimal route for a delivery vehicle. Consequently, the corresponding mathematical formulation can be rigorously expressed as follows.

2.2.1. Objective Function

The purpose of TSP is to minimize all selective paths in order to identify the optimum route for a delivery vehicle.

$$\mathit{min}TSP f\left(y\right)=\sum _{j=1}^n·\sum _{k=1}^n\left(y_{jk}{d}_{jk}\right)$$

(1)

2.2.2. Constraints

The transport vehicle has an ability to deliver goods to each international destination for each delivery only once.

$$\sum _{j=1}^n\left(y_{jk}\right)= 1 \forall k \in \{1,2,3,\dots n\}$$

(2)

The transport vehicle has an ability to leave each international destination for each delivery only once.

$$\sum _{k=1}^n\left(y_{jk}\right)= 1 \forall j \in \{1,2,3,\dots n\}$$

(3)

The transport vehicle does not exhibit the sub contour activity.

$$\sum _{j,k\in S}^n\left(y_{jk}\right)\le \left|S\right|-1 S\subset V, 2\le |S|\le n-2$$

(4)

The decision variable governing the selection of the transportation routes.

$$y_{jk}\in \left\{\mathrm{1,0}\right\} j, k\in \{\mathrm{1,2},3,\dots \mathrm{n}\}$$

(5)

where

$y_{jk}$ = 1 The driver selectively drives vehicle from location j to location k.

$y_{jk}$ = 0 The driver unselectively drives vehicle from location j to location k.

2.3. Problem Formulation

In international land logistics, service inefficiency is often assessed through transportation cost minimization. Consolidating bulky goods at the country of origin and moving them across borders over long distances imposes substantial financial burdens. Yet transport cost depends not only on distance but also on time, which significantly influences international road freight expenses. Fuel consumption, for example, is affected by mileage as well as delays caused by congestion, accidents, and roadworks, all of which increase duration without adding value. From a symmetry perspective in supply chain design, distance and time should be treated as complementary variables. Accordingly, their joint consideration enables more accurate freight planning and total cost estimation.

Efficient international road transportation depends on three interrelated factors: distance, time, and cost, with transport cost shaped by both spatial and temporal variables. Although the classical Traveling Salesman Problem (TSP) has been extensively studied, it typically prioritizes distance alone (TTSPD), leaving a gap in frameworks that jointly model time and distance. To address this limitation, we extend the TSP to an Extended Traveling Salesman Problem (ETSPDT), which explicitly incorporates both dimensions and reformulates the objective function to capture fuel consumption. By linking energy use to transportation cost, this framework enables a more rigorous assessment of delivery efficiency. Within supply chain and logistics management, such models support the analysis of symmetry in route structure and operational balance, offering valuable insights for optimizing container truck delivery systems under real-world constraints.

2.3.1. Traditional Traveling Salesman Problem in Term of Distance Concentration: TTSPD

In the context of international road transport for cross-border product distribution, the classical objective has been to identify a route that minimizes the total travel distance per tour. The process entails dispatching shipments from the supplier in the origin country, consolidating them, and transporting them across multiple destinations before returning to the point of origin. This operational problem can be rigorously formalized as TTSPD, in which historical transport distances serve as the primary input for estimating associated costs. Accordingly, the TTSPD admits a well-defined mathematical formulation coupled with having notations of additional parameters as in

Table 2. Notations of additional parameters in TTSPD computation. Source: the Authors.

Notations	Description
$v_{jk}$	Velocity which vehicle can perform from location j to $k.$
$e_{jk}$	Fuel consumption rate from location j to $k,$ depended on $v_{jk}.$
$TTSPD\left(y\right)$	Total transport distance in one trip following the concept of Traditional Traveling Salesman Problem based on distance.
$Transport cost TTSPD\left(y\right)$	The total cost in one trip following the concept of Traditional Traveling Salesman Problem based on distance.

, which provides a foundational framework for subsequent cost optimization analyses.

The TTSPD paradigm is fundamentally predicated upon the spatial distances traversed within a product delivery itinerary. It is conventionally employed to allocate goods across a network of terminal nodes or points of consumption. The central aim of this framework lies in the minimization of aggregate travel distance, thereby delineating the most efficient routing scheme. Crucially, the optimization of such routes is inherently subordinate to the overarching objective of curtailing transportation costs, with particular emphasis on a reduction in vehicular fuel expenditure. In this regard, fuel consumption exhibits a direct dependence on vehicle velocity, which is itself contingent upon the driver’s proficiency in negotiating inter-node transitions. Consequently, the mathematical formulation of the resulting model can be expressed in the following formal terms.

Objective Function

The primary objective function is to indicate the shortest part.

$$\mathit{min}TTSPD f\left(y\right)=\sum _{j=1}^n·\sum _{k=1}^n\left(y_{jk}{d}_{jk}\right)$$

(6)

The subsequent objective function is the consequence from the primary objective function, since each travel from location j to location j might let the driver perform the dissimilar velocity ($v_{jk}$), which will subsequently indicate the different energy consumption rate. In addition, transport cost consequently is the summation of each energy consumption multiplying with each distance in the designed route.

$$\mathit{Transport\; cost} TTSPD f\left(y\right)=\sum _{j=1}^n·\sum _{k=1}^n\left(e_{jk}{y}_{jk}{d}_{jk}\right)$$

(7)

Constraints

The transport vehicle only has one time to visit each international destination.

$$\sum _{j=1}^n\left(y_{jk}\right)= 1 \forall k \in \{1,2,3,\dots n\}$$

(8)

The transport vehicle only has one time to leave each international destination.

$$\sum _{k=1}^n\left(y_{jk}\right)= 1 \forall j \in \{1,2,3,\dots n\}$$

(9)

The next path is primarily selected under the shortest transport distance form.

$$\sum _{j=1}^n\left(d_{jk}\right)\left(y_{jk}\right)=min \left(d_{jk}\right) \forall k \in \{1,2,3,\dots n\}$$

(10)

The transport vehicle does not have the activity of the sub contour.

$$\sum _{j,k\in S}^n\left(y_{jk}\right)\le \left|S\right|-1 S\subset V, 2\le |S|\le n-2$$

(11)

The decision variable for selecting the transport path.

$$y_{jk} \in \left\{\mathrm{1,0}\right\} j, k \in \{\mathrm{1,2},3,\dots n\}$$

(12)

where

$y_{jk}$ = 1 The driver selectively drives vehicle from location j to location k.

$y_{jk}$ = 0 The driver unselectively drives vehicle from location j to location k.

Overall, the TTSPD formulation is a discrete binary integer optimization model derived from the classical traveling salesman problem. It is single-objective when either the total transport distance or transport cost is minimized independently; however, it can be extended to a multi-objective framework when both criteria are optimized simultaneously. Due to the binary routing decisions and subtour-elimination constraints, the model is nonconvex. Moreover, it remains linear if fuel consumption is treated as a constant parameter but becomes nonlinear when fuel consumption is modeled as a function of vehicle velocity. For problems involving a large number of destinations, the model is also large-scale and therefore computationally challenging.

2.3.2. Extended Travelling Salesman Problems Based in Term of Distance and Time Concentration: ETSPDT

This study addresses the computational solution of the Closed-Loop Traveling Salesman Problem (CLTSP), a combinatorial optimization problem that identifies the most efficient cyclic route from a given origin to multiple destinations, subject to the constraint that each location is visited exactly once before returning to the departure point. In international road freight logistics, this problem becomes especially complex, as vehicles must leave a domestic terminal, cross national borders, and deliver goods sequentially to geographically dispersed locations while preserving the closed-loop structure. From a symmetry perspective, the route exhibits a circular configuration in which the origin and destination nodes are treated within a unified spatial framework. Exploiting such symmetry can improve route evaluation and solution efficiency. Therefore, developing optimal CLTSP solutions is essential for enhancing operational performance and reducing transportation costs in transnational supply chains.

In real-world logistics, the shortest route is not always the fastest. Flooding, congestion, accidents, and other exogenous disruptions can increase travel time even on geographically shorter paths. Accordingly, route optimization should account for both spatial and temporal factors. To this end, the present study extends the classical model by integrating distance and time into a unified mathematical framework, yielding the ETSPDT. The model incorporates additional parameters, as summarized in

Table 3. Notations of additional parameters in ETSPDT computation. Source: the Authors.

Notations	Description
${lt}_{jk}$	Transport time which the transport vehicle can perform from location j to location $k.$
$ETSPDT\left(y\right)$	The rate of the total transport time per total transport distance, which explains CLTSP in the international road freight context.
$Transport cost ETSPDT\left(y\right)$	The total cost in one trip following the concept of Extended Travelling Salesman Problems based on distance and time (ETSPDT).

, and its theoretical basis and formal definition are presented in the following sections. From a symmetry perspective, this formulation preserves structural balance while capturing asymmetric operational conditions, thereby offering a robust basis for advanced supply chain and logistics applications.

The primary objective of the ETSPDT is to determine an optimal routing strategy for each agent, encompassing the transportation of goods across multiple international destinations and the subsequent return to the domestic origin. This optimization problem seeks to minimize the cumulative travel time in proportion to the total distance traversed, while concurrently incorporating the vehicle’s energy consumption profile as a critical operational constraint. In essence, the task is reduced to identifying the route that simultaneously maximizes temporal efficiency and energy economy, a goal that can be rigorously formalized through the subsequent mathematical problem formulation.

Objective Function

The primary objective function is to indicate the fastest movement when considering transport time and transport distance at the same time.

$$\mathit{min}ETSPDT f\left(y\right)={\displaystyle \frac{{\sum }_{j=1}^n·{\sum }_{k=1}^n\left(y_{jk}{lt}_{jk}\right)}{{\sum }_{j=1}^n·{\sum }_{k=1}^n\left(y_{jk}{d}_{jk}\right)}}$$

(13)

The second objective function is the minimization from the primary objective function, since each transport time (${lt}_{jk}$) coupled with distancing ($d_{jk}$) might let the driver perform the dissimilar velocity ($v_{jk}$), which will indicate the different energy consumption rate. In addition, transport cost consequently is the summation of each energy consumption multiplying with each distance in the designed route.

$$\mathit{min\; Transport\; cost} ETSPDT f\left(y\right)=\sum _{j=1}^n·\sum _{k=1}^n\left(e_{jk}{y}_{jk}{d}_{jk}\right)$$

(14)

Constraints

The transport vehicle only has one time to visit each international destination.

$$\sum _{j=1}^n\left(y_{jk}\right)= 1 \forall k \in \{1,2,3,\dots n\}$$

(15)

The transport vehicle only has one time to leave each international destination.

$$\sum _{k=1}^n\left(y_{jk}\right)= 1 \forall j \in \{1,2,3,\dots n\}$$

(16)

The next path is primarily selected under the condition of the lowest fuel consumption based on each transport time per transport distance, which will be able to be transformed to a dissimilar velocity.

$$\sum _{j=1}^n\left(e_{jk}{y}_{jk}{d}_{jk}\right)=min \left(e_{jk}{y}_{jk}{d}_{jk}\right) \forall k \in \{1,2,3,\dots n\}$$

(17)

where $e_{jk}$. The fuel consumption depends on $v_{jk}$ = 1/(${lt}_{jk}/d_{jk}$).

The transport vehicle does not have the activity of the sub contour.

$$\sum _{j,k\in S}^n\left(y_{jk}\right)\le \left|S\right|-1 S\subset V, 2\le |S|\le n-2$$

(18)

The decision variable is for selecting the transport path.

$$y_{jk} \in \left\{\mathrm{1,0}\right\} j, k \in \{\mathrm{1,2},3,\dots n\}$$

(19)

where

$y_{jk}$ = 1 The driver selectively drives vehicle from location j to location k.

$y_{jk}$ = 0 The driver unselectively drives vehicle from location j to location k.

Equations (13)–(19) underpin the derivation of globally optimal transportation routes, whereby each transport agent executes a time-minimizing trajectory from its domestic parking facility, transits international borders to deliver goods across multiple foreign destinations, and ultimately returns to its point of origin, thereby ensuring maximal operational efficiency and adherence to logistical constraints.

On the whole, the proposed ETSPDT formulation is a deterministic, nonlinear, nonconvex binary integer optimization model. Structurally, it belongs to the class of extended travelling salesman problems because it imposes standard routing requirements, including a single visit and single departure for each destination, together with subtour elimination constraints. The model is primarily single-objective, with minimization of the total transport-time-to-distance ratio serving as the main criterion, while transport cost is introduced as an additional operational performance measure associated with energy consumption. Since the decision variables are binary and the number of route-selection variables grows quadratically with the number of destinations, the model is combinatorial in nature and may become large-scale for practical instances involving many international locations.

2.4. The Accurate Distance Between Two Locations

From antiquity to the present, scholars and practitioners have persistently sought to devise systematic methods for quantifying the spatial separation between distinct locations. These endeavors frequently rely on computational algorithms that determine distances from the latitudinal and longitudinal coordinates of the points under consideration. A canonical example is the application of the Pythagorean theorem, which yields the length of the side opposite the right angle in a triangle commonly referred to as the Euclidean distance [40,41]. Yet, when distances are measured along a great circle, the geodesic that constitutes the shortest path between two points on a sphere planar approximations become inadequate. This approach explicitly accounts for the non-uniformity of latitude and longitude by incorporating the Earth’s curvature, typically through the Haversine formula, which employs trigonometric functions to adjust for the planet’s spherical geometry. In this framework, therefore, a “straight line” is more accurately conceptualized as a curved arc [42,43]. While such methods provide a meaningful approximation of inter-location distances, they remain inherently approximate, as they do not capture the true travel distance.

In international land logistics, conventional distance metrics such as Euclidean, great-circle, and Haversine measures quantify spatial separation but fail to capture the constraints of real-world road networks. Reliance on such idealized straight-line distances can produce suboptimal or hazardous transport routes, intersecting agricultural, horticultural, or urban areas. Consequently, classical distance-based approaches are insufficient for decision-support models, necessitating frameworks that integrate network topology, road hierarchies, and vehicular constraints to enable mathematically rigorous optimization of cross-border goods distribution.

At present, the application Google Maps facilitates the precise quantification of geospatial distances between arbitrary locations. Functioning as an advanced computational tool, it enables optimized navigation by determining the minimal-cost path connecting any two points on the Earth’s surface. The platform, developed by Google Inc., represents the spatial network as a graph-based data structure and employs Dijkstra’s algorithm to systematically identify the globally shortest path. This algorithm rigorously evaluates all feasible trajectories, iteratively updating tentative distances to converge on the path that minimizes cumulative traversal cost. In parallel, Google Maps assimilates real-time GPS information from active users to model prevailing traffic dynamics, thereby providing refined estimates of expected travel time [44,45,46]. Consequently, the system embodies a synthesis of classical graph-theoretic optimization and contemporary data-driven analytics, ensuring both computational efficiency and practical navigational accuracy.

Moreover, a systematic analysis of multiple geospatial points enables the precise quantification of inter-location distances and robust estimation of travel times, effectively positioning Google Maps as a real-time traffic analytics platform. Integrating such dynamic data into computational algorithms is expected to enhance the fidelity of transportation optimization models. Accordingly, the present study employs a methodology that computes distances and travel durations across heterogeneous transportation nodes by leveraging real-time streams from the Google Maps database, ensuring both computational rigor and empirical relevance.

2.5. Conceptual Framework

The Traveling Salesman Problem (TSP) has traditionally been framed as a distance-minimization model, in which the shortest route among multiple destinations is assumed to reduce transportation cost and energy consumption. However, this symmetry between distance and efficiency is often disrupted in real logistics systems by traffic congestion, border delays, and road disruptions, where stationary vehicles continue to consume fuel and time. To address this limitation, the present study first considers the Traditional TSP defined by distance (TTSPD) as a baseline and then develops an Extended TSP incorporating Distance and Time (ETSPDT). This dual-parameter formulation captures both spatial and temporal dimensions, thereby reflecting the asymmetric realities of international road freight transport. Initially, TTSPD was the first model derived from the fundamental concept of the traditional TSP, which primarily focuses on path distance before route selection. It was employed to validate ESTPDT, which extends this framework by considering distance and time simultaneously, thereby serving as the driving development model in this research. Expectedly, comparative results indicate that ETSPDT more effectively represents energy-use dynamics and offers a stronger theoretical foundation for cross-border logistics optimization. Accordingly, the model supports more accurate decisions aimed at reducing energy consumption and transport cost. Figure 1 illustrates this progression from classical distance-based optimization to a symmetry-aware, dual-parameter approach.

3. Research Method

3.1. Initial Population

This study assesses the cost-effectiveness of transportation by comparing vehicle energy consumption between the TTSPD and ETSPDT models. Direct measurement of actual travel distances and durations in real-world conditions is challenging, primarily due to substantial financial and logistical constraints. As an alternative, Google Maps was employed, leveraging aggregated real-time GPS data from users to provide accurate distance estimates and reliable travel time predictions under prevailing traffic conditions. International road freight destinations were randomly selected using Google Maps, with initial departure points in Thailand similarly determined from the Administrative Report [47]. Cross-border districts within ASEAN countries with established land connectivity specifically Cambodia, the Lao People’s Democratic Republic, Myanmar, and Vietnam (CLMV nations) were also identified randomly. All geographic references were standardized using city or district names, omitting specific organizations.

Consistent with quantitative methodological standards, particularly the “rule of thumb” principle [48], the study ensured that the number of transport paths exceeded thirty. For each route, comprehensive data, including distance $d_{jk}$ and estimated travel time $t_{jk}$, were systematically collected. This approach integrates real-time geospatial information with rigorous sampling to approximate actual transport distances, durations, and traffic conditions, providing a reliable foundation for subsequent energy consumption and cost-effectiveness analyses.

3.2. Vehicle Energy Consumption Rate

Within the methodological frameworks employed in this study, vehicle energy consumption emerges as a direct consequence of the objective functions defined in each mathematical model, thereby providing a quantitative basis for evaluating transportation efficiency. This modeling perspective aligns with real-world logistics, where trucks represent the predominant means of international road freight, facilitating the cross-border movement of goods and reflecting the operational realities captured by the models.

Recently, ref. [49] investigated fuel consumption and its determinants in three-axle container trucks, identifying road slope (S, %), vehicle velocity (V, km/h), and total vehicle weight (W, tons) as the primary factors. Based on these variables, they developed predictive fuel consumption models (C, L/km) for two operational scenarios.

In the first scenario (W: 11.7–28 t, S: 2–7%, V > 50 km/h), fuel consumption was described by C = (105.635S − 3.64516V − 8.00992W + 15.2035SW − 20.5096S² − 0.0270028S²W²)/1000 while the second scenario (W: 11.7–28 t, S: 0.6–2%, V > 50 km/h), the model was C = (38.8019S + 2.47673V + 14.595W − 0.207869VW − 1.24498SV + 0.135391SVW)/1000.

To apply this framework to international road freight, an average truck load was considered, reflecting that vehicles are fully loaded only at the origin and progressively unload en route, arriving empty at the final destination. Additionally, mean slope values were employed to represent route gradients, acknowledging variability along the transport path. These parameters were subsequently integrated into the analysis, providing a practical basis for evaluating energy consumption across international freight routes as outlined below.

The energy consumption rate for the truck contains the product weight 19.85 tones.

Stage 1 The transport condition: slope: 4.5%, and velocity below 50 km/h.

$$e_{jk}=(1043.63968-3.64516v_{jk})/1000$$

(20)

Stage 2 The transport condition: slope: 1.3%, and velocity above 50 km/h.

$$e_{jk}=(340.15322-3.13255v_{jk})/1000$$

(21)

The TTSPD approach was initially formulated to determine the shortest path, from which the corresponding transport time for each route was derived. Path velocity ($v_{jk}$) was then computed as the ratio of distance to transport time. By contrast, the ETSPDT method prioritized time efficiency by minimizing transport time per unit distance, with path selection guided by the inverse of minimal fuel energy consumption ($e_{jk})$ per distance, serving as a proxy for ($v_{jk}$). For both frameworks, velocities were categorized according to speed thresholds—for example, below or above 50 km/h for container truck transport. One of Equations (20) and (21) was subsequently applied to determine the energy consumption rate ($e_{jk}$) for each path, which then fed into Equations (7) and (13) to assess cost-effectiveness under TTSPD and ETSPDT, respectively.

In essence, Equations (20) and (21) provide the basis for evaluating cost-effectiveness, facilitating the optimization of international truck freight. Objective functions were first converted into path velocities, which were then used to calculate energy consumption and total fuel usage, ultimately yielding the transportation costs associated with each logistical model.

3.3. Data Analysis

The data employed in this study were primarily subjected to quantitative analysis. Initially, the dataset comprising various visited locations was randomly obtained through the Google Maps application. These data points were then systematically analyzed to assess cost effectiveness based on the theoretical frameworks of the TTSPD and ETSPDT. The outcomes of these exploratory analyses yielded novel insights that may serve as practical guidelines for optimizing international logistics operations specifically by reducing fuel consumption in cross-border road freight transportation. Furthermore, the comprehensive data analysis was conducted using the Solver Add-in tool within Microsoft Excel 2024. The two mathematical models grounded in the TTSPD and ETSPDT concepts were subjected to numerical validation through the procedures that are exhibited in Figure 2 and Figure 3, respectively. Moreover, TTSPD and ETSPDT can be solved by applying linear programming algorithms, as illustrated in Algorithms 1 and Algorithm 2 below.

Algorithm 1 Algorithm Solve TTSPD by Linear Programing Solver

Input: $d_{jk}$, $e_{jk}$, $v_{jk}$ $, y_{jk}, n$ Output:   Optimal route to indicate the shortest part ${\sum }_{j=1}^n.{\sum }_{k=1}^n\left(y_{jk}{d}_{jk}\right)$,   Transport Cost ${\sum }_{j=1}^n.{\sum }_{k=1}^n\left(e_{jk}{y}_{jk}{d}_{jk}\right)$ 1: Read data $d_{jk}$, $e_{jk}$, $n$ 2: Define the binary decision variable $y_{jk}$    $y_{jk}$ = 1 The driver selectively drives vehicle from the location j to the location k.   $y_{jk}$ = 0 The driver unselectively drives vehicle from the location j to the location k. 3: Define objective function to minimize total transport distance: ${\sum }_{j=1}^n·{\sum }_{k=1}^n\left(y_{jk}{d}_{jk}\right)$ 4: Add visit-once constraints: ${\sum }_{j=1}^n\left(y_{jk}\right)= 1 , \forall \mathrm{k}$ 5. Add leave-once constraints: ${\sum }_{k=1}^n\left(y_{jk}\right)= 1 , \forall \mathrm{j}$ 6. Add subtour elimination constraints: ${\sum }_{j,k\in S}^n\left(y_{jk}\right)\le \left|\mathrm{S}\right|-1,\mathrm{S}\subset \mathrm{V},2\le \left|\mathrm{S}\right|\le \mathrm{n}-2$ 7. Call Linear Programing Solver 8. Extract selected arcs where $y_{jk}$ = 1 9. Compute   Total Distance ${\sum }_{j=1}^n·{\sum }_{k=1}^n\left(y_{jk}{d}_{jk}\right)$ and   Subsequent Transport Cost ${\sum }_{j=1}^n·{\sum }_{k=1}^n\left(e_{jk}{y}_{jk}{d}_{jk}\right)$ 10. Return optimal route by concentrating on distance End Algorithm.

Algorithm 2 Algorithm Solve ETSPDT by Linear Programing Solver

Input: $d_{jk}$, $e_{jk}$, $v_{jk}, y_{jk}, n$ Output:   Optimal route to indicate the fastest movement by considering transport time and distance at the same time: ${\displaystyle \frac{{\sum }_{j=1}^n·{\sum }_{k=1}^n\left(y_{jk}{lt}_{jk}\right)}{{\sum }_{j=1}^n·{\sum }_{k=1}^n\left(y_{jk}{d}_{jk}\right)}}$,   Transport Cost: ${\sum }_{j=1}^n·{\sum }_{k=1}^n\left(e_{jk}{y}_{jk}{d}_{jk}\right)$ 1: Read data $d_{jk}$, $e_{jk}$, $n$ 2: Define the binary decision variable $y_{jk}$    $y_{jk}$ = 1 The driver selectively drives vehicle from the location j to the location k.   $y_{jk}$ = 0 The driver unselectively drives vehicle from the location j to the location k. 3: Define objective function to minimize total transport distance: ${\sum }_{j=1}^n·{\sum }_{k=1}^n\left(y_{jk}{d}_{jk}\right)$ 4: Add visit-once constraints: ${\sum }_{j=1}^n\left(y_{jk}\right)= 1 , \forall \mathrm{k}$ 5. Add leave-once constraints: ${\sum }_{k=1}^n\left(y_{jk}\right)= 1 , \forall \mathrm{j}$ 6. Add the next path constraints: $v_{jk}$= 1/(${lt}_{jk}/d_{jk}$) 7. Add subtour elimination constraints: ${\sum }_{j,k\in S}^n\left(y_{jk}\right)\le \left|\mathrm{S}\right|-1,\mathrm{S}\subset \mathrm{V},2\le |\mathrm{S}|\le \mathrm{n}-2$ 8. Call Linear Programing Solver 9. Extract selected arcs where $y_{jk}$ = 1 10. Compute   Total Distance ${\displaystyle \frac{{\sum }_{j=1}^n·{\sum }_{k=1}^n\left(y_{jk}{lt}_{jk}\right)}{{\sum }_{j=1}^n·{\sum }_{k=1}^n\left(y_{jk}{d}_{jk}\right)}}$   Transport Cost ${\sum }_{j=1}^n·{\sum }_{k=1}^n\left(e_{jk}{y}_{jk}{d}_{jk}\right)$ 11. Return optimal route by considering transport time and distance at the same time End Algorithm.

4. Results

4.1. Initial Data

This study employed a probabilistic random sampling framework to identify points of origin based on administrative datasets from Thailand, in conjunction with corresponding international delivery destinations in the Lao People’s Democratic Republic (Lao PDR) and Cambodia (n), as geospatially resolved via the Google Maps platform. A total of 23 delivery nodes, including the primary departure terminal, were systematically incorporated into the analysis. The delivery trajectory was initiated from Ubon Ratchathani, Thailand, from which consignments were stochastically allocated to 10 destinations in Lao PDR and 12 in Cambodia, respectively. Each origin and destination node, represented by precise intercontinental coordinates, was rigorously tabulated and subsequently visualized to facilitate comprehensive spatial analysis, as presented in

Table 4. The random visiting cities Source: the Authors.

No.	City	Country	Position
1	Ubon Ratchathani	Thailand	Starting point: Loading point
2	Pakse	Lao PDR	International Destination
3	Sanasomboun	Lao PDR	International Destination
4	Batiengchaleunsouk	Lao PDR	International Destination
5	Paksong	Lao PDR	International Destination
6	Pathouphone	Lao PDR	International Destination
7	Phonthong	Lao PDR	International Destination
8	Ban Fang Deng	Lao PDR	International Destination
9	Soukhoumma	Lao PDR	International Destination
10	Ban None Champa	Lao PDR	International Destination
11	Ban Vangtao Nok	Lao PDR	International Destination
12	Tboung Khmum	Cambodia	International Destination
13	Battambang	Cambodia	International Destination
14	Preah Sihanouk	Cambodia	International Destination
15	Phnom Penh	Cambodia	International Destination
16	Kandal	Cambodia	International Destination
17	Stung Treng	Cambodia	International Destination
18	Kampong Speu	Cambodia	International Destination
19	Svay Rieng	Cambodia	International Destination
20	Kampong Chhnang	Cambodia	International Destination
21	Prey Veng	Cambodia	International Destination
22	Kampot	Cambodia	International Destination
23	Pailin	Cambodia	International Destination

.

As summarized in Table 4, the present study considered a set of 23 essential locations distributed across three countries, each requiring visitation. Consequently, the total number of candidate transport routes was 529 tracks, corresponding to all ordered pairs of origin and destination cities.

Inter-location distances were systematically obtained using the Google Maps platform, from which a comprehensive symmetrical distance matrix ($d_{jk}$) was constructed, representing feasible vehicular travel distances in kilometers (km), as detailed in

Table 5. Symmetrical Distance matrix (km) for road line transport. Source: the Authors.

(j, k)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23
1	0	139	178	171	188	205	120	125	140	125	103	504	418	742	544	619	327	582	629	572	560	675	464
2	136	0	39.1	31.4	49	65.8	19.1	20.1	85.9	19.5	38.4	462	525	790	592	640	223	630	549	619	524	723	571
3	175	39.1	0	70.5	88	105	58.1	59.2	125	58.6	77.4	501	564	830	631	679	262	669	588	659	563	762	610
4	168	31.4	70.5	0	42	83.7	50.5	51.5	117	50.9	69.8	480	556	808	610	658	240	648	567	637	542	740	602
5	185	48.9	88	42	0	101	68	69	135	68.4	87.3	498	574	826	627	676	258	665	585	655	559	758	620
6	202	65.9	105	83.7	101	0	84.5	85.5	98.9	84.9	104	436	652	764	565	614	196	603	523	593	497	696	698
7	117	19.1	58.2	50.5	68	84.4	0	4.9	95.6	4.7	19.3	481	506	809	610	659	241	649	568	638	542	741	552
8	122	20.1	59.2	51.5	69	85.5	4.9	0	96.6	1.5	24.1	482	511	810	611	660	242	650	569	639	543	742	557
9	140	85.8	125	117	135	98.9	95.5	96.6	0	96	115	429	480	757	558	607	189	597	516	586	490	689	526
10	122	19.5	58.6	50.9	68.4	84.9	4.7	1.5	96	0	24	481	511	809	611	659	242	649	568	638	543	742	557
11	99.9	38.3	77.5	69.8	87.3	104	19.3	24.1	115	24	0	500	489	828	630	678	260	668	587	657	562	760	535
12	505	462	501	480	498	436	481	482	429	481	500	0	366	336	131	180	246	175	126	170	63.1	274	440
13	417	524	563	556	573	651	505	510	481	510	488	367	0	469	290	380	459	307	411	202	356	406	77.5
14	743	791	830	808	826	764	809	810	757	810	828	336	468	0	215	289	575	166	345	273	294	101	542
15	544	592	631	609	627	565	610	611	558	611	629	131	290	215	0	75.1	376	54.2	122	94.1	91.2	152	364
16	619	639	678	657	675	613	658	659	606	658	677	179	365	289	75.4	0	423	129	149	169	118	164	439
17	327	223	262	240	258	196	241	242	189	242	260	246	459	575	376	424	0	414	334	404	308	507	505
18	582	630	669	648	665	603	649	650	597	649	668	175	306	166	54.2	129	414	0	184	111	133	123	381
19	629	549	589	567	585	523	568	569	516	569	587	126	410	345	122	149	334	185	0	215	80.6	261	485
20	623	620	659	638	656	594	639	640	587	639	658	171	202	273	94	184	404	111	215	0	160	210	276
21	561	524	563	541	559	497	542	543	490	543	561	63.1	355	294	91.4	118	308	134	80.9	159	0	230	429
22	676	723	762	741	759	697	742	743	690	742	761	274	406	101	152	164	507	123	261	210	230	0	480
23	476	583	622	615	632	710	564	569	540	569	547	443	78.9	545	366	456	518	383	487	278	432	482	0

. The minimal observed route length was 1.5 km, corresponding to the segment connecting Ban Fang Deng and Ban None Champa in Lao PDR, whereas the maximal route extended 830 km, linking Paksong in Lao PDR to Preah Sihanouk in Cambodia. It is noteworthy that the distances associated with outbound and return journeys were not always symmetric, reflecting Google Maps’ integration of dynamic traffic conditions and potential directional or accessibility constraints inherent to specific road segments. Hence, the derived distance matrix not only encapsulates contemporaneous traffic patterns but also constitutes a rigorously defined framework for the determination of accurate and operationally viable transport pathways.

4.2. Delivery Based on TTSPD

To determine the most efficient route, the shortest distances were analyzed from the initial departure point through multiple international destinations, culminating at the final terminal and returning to the origin. The optimal path $d_{jk}$ was selected via the decision variable $y_{jk}$ following the traditional distance-based transportation problem framework illustrated in Figure 2. The resulting shortest route and corresponding vehicle movements are summarized in

Table 6. The path for transporting following TTSPD. Source: the Authors.

Order	j to k	Distance (km): ${\mathit{y}}_{\mathit{j}\mathit{k}}{\mathit{d}}_{\mathit{j}\mathit{k}}$	Time (h)	Velocity (km/h): ${\mathit{v}}_{\mathit{j}\mathit{k}}$	Fuel Consumption Rate (L/km): ${\mathit{e}}_{\mathit{j}\mathit{k}}$ Stage 1	Fuel Consumption Rate (L/km): ${\mathit{e}}_{\mathit{j}\mathit{k}}$ Stage 2	Amount (L): ${\mathit{e}}_{\mathit{j}\mathit{k}}{\mathit{y}}_{\mathit{j}\mathit{k}}{\mathit{d}}_{\mathit{j}\mathit{k}}$
1	1–11	103.0	1.7	61		0.15	15.3
2	11–7	19.3	0.4	55		0.17	3.2
3	7–10	4.7	0.1	36	0.91		4.3
4	10–8	1.5	0.1	19	0.98		1.5
5	8–2	20.1	0.5	44	0.88		17.8
6	2–4	31.4	0.7	48	0.87		27.2
7	4–5	42.0	0.8	56		0.16	6.9
8	5–3	88.0	1.6	57		0.16	14.3
9	3–6	105.0	2.2	47	0.87		91.6
10	6–9	98.9	3.0	33	0.92		91.3
11	9–17	189.0	4.3	44	0.88		167.2
12	17–12	246.0	4.1	61		0.15	36.9
13	12–21	63.1	1.2	52		0.18	11.2
14	21–19	80.9	1.6	50	0.86		69.8
15	19–15	122.0	2.5	49	0.87		105.7
16	15–18	54.2	1.3	42	0.89		48.3
17	18–20	111.0	2.1	53		0.18	19.5
18	20–16	184.0	3.7	49	0.86		158.9
19	16–22	164.0	3.2	51		0.18	29.5
20	22–14	101.0	2.3	43	0.89		89.4
21	14–13	468.0	6.2	76		0.10	47.8
22	13–23	77.5	1.6	49	0.86		67.0
23	23–1	476.0	7.6	62		0.14	68.9
	Total	2850.6	52.7				1193.5

.

As seen in Table 6, the journey began in Ubon Ratchathani, Thailand, proceeded through Laos PDR—visiting Ban Vangtao Nok, Phonthong, Ban None Champa, Ban Fang Deng, Pakse, Batiengchaleunsouk, Paksong, Sanasomboun, Pathouphone, and Soukhoumma—and then entered Cambodia, traversing Stung Treng, Tboung Khmum, Prey Veng, Svay Rieng, Phnom Penh, Kampong Speu, Kampong Chhnang, Kandal, Kampot, Sihanoukville, Battambang, and Pailin, before reaching Preah Sihanouk. The route concluded with a return to Ubon Ratchathani. Distances and transportation orders were derived according to the TTSPD methodology.

As shown in Table 6, the functions of TTSPD were determined. Originally, the shortest route minimization was the primary objective function for this scenario of international delivery, which was 2850.6 km coupled with using the total transport time 52.7 h. In addition, the fuel consumption was its cost effectiveness function, which in total was predicted to be 1193.5 L regarding carrying by truck.

4.3. Delivery Based on ETSPDT

To determine the optimal minimization of the total travel time per unit of transport distance, in conjunction with fuel consumption, the analysis simultaneously considered temporal, spatial, and energetic factors prior to route selection. The itinerary was structured to originate from the home location, proceed sequentially through designated international destinations, reach the final foreign nodes, and subsequently return to the initial departure point. Prior to evaluating the total travel time normalized by transport distance, it was necessary to construct and analyze the symmetrical time matrix, which encodes the travel durations between each pair of locations i and j. This symmetrical matrix was generated using the Google Maps platform, and the resulting data are summarized in

Table 7. Symmetrical Time matrix (h) for road line transport. Source: the Authors.

(j, k)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23
1	0	2.3	3	2.93	3.16	3.81	1.93	2.1	3.1	2.08	1.68	8.61	6.38	11.05	9.08	11.01	5.63	9.58	10.85	9.2	9.63	11.23	7.51
2	2.3	0	0.7	0.65	0.86	1.53	0.36	0.48	2.06	0.43	0.73	7.41	8.15	11.75	9.78	11.18	3.5	10.3	8.85	9.91	8.55	11.93	9.28
3	2.98	0.7	0	1.35	1.56	2.23	1.05	1.16	2.75	1.11	1.41	8.11	8.83	12.45	10.48	11.88	4.2	11	9.55	10.61	9.25	12.63	9.96
4	2.91	0.63	1.33	0	0.75	1.86	1	1.11	2.7	1.06	1.36	7.75	8.78	12.08	10.11	11.51	3.83	10.63	9.18	10.25	8.88	12.26	9.9
5	3.13	0.85	1.55	0.71	0	2.06	1.2	1.31	2.9	1.26	1.56	7.95	8.98	12.28	10.31	11.71	4.03	10.83	9.38	10.45	9.08	12.46	10.1
6	3.83	1.56	2.26	1.88	2.11	0	1.9	2.01	3	1.96	2.26	7.21	11.01	11.55	9.58	10.98	3.3	10.1	8.65	9.71	8.35	11.73	12.13
7	1.93	0.35	1.06	1	1.23	1.88	0	0.16	2.16	0.13	0.36	7.75	7.78	12.08	10.11	11.51	3.83	10.63	9.18	10.25	8.88	12.28	8.9
8	2.11	0.46	1.16	1.1	1.33	1.98	0.18	0	2.28	0.08	0.55	7.85	7.96	12.2	10.23	11.63	3.93	10.73	9.3	10.36	8.98	12.38	9.08
9	3.11	2.08	2.78	2.71	2.95	3.03	2.2	2.31	0	2.26	2.56	8.25	8.15	12.58	10.63	12.01	4.33	11.13	9.68	10.75	9.38	12.78	9.26
10	2.08	0.45	1.15	1.08	1.31	1.96	0.16	0.08	2.26	0	0.51	7.83	7.95	12.18	10.21	11.61	3.91	10.71	9.26	10.33	8.96	12.36	9.06
11	1.7	0.71	1.41	1.35	1.58	2.23	0.35	0.51	2.53	0.5	0	8.1	7.55	12.45	10.48	11.88	4.18	10.98	9.53	10.61	9.23	12.63	8.66
12	8.58	7.51	8.21	7.85	8.06	7.28	7.86	7.98	8.3	7.93	8.23	0	5.73	4.81	2.75	3.86	4.1	3.36	2.33	3.23	1.21	5.25	7.1
13	6.43	8.18	8.9	8.83	9.06	11.06	7.83	8	8.18	7.96	7.58	5.96	0	6.3	4.53	6.41	7.78	4.78	7.06	2.96	5.63	6.4	1.58
14	11.06	11.88	12.58	12.2	12.43	11.63	12.21	12.33	12.65	12.28	12.58	4.83	6.16	0	2.8	4.13	8.45	1.95	4.95	3.65	4.21	2.43	7.53
15	9.03	9.85	10.55	10.16	10.4	9.61	10.18	10.3	10.61	10.26	10.55	2.7	4.3	2.75	0	1.93	6.41	1.3	2.56	1.8	2.03	2.9	5.66
16	10.98	11.23	11.93	11.55	11.78	11	11.58	11.7	12.01	11.65	11.93	3.8	6.25	4.15	1.96	0	7.81	2.68	3.25	3.75	2.7	3.21	7.63
17	5.58	3.55	4.25	3.88	4.1	3.31	3.9	4.01	4.33	3.96	4.26	4.05	7.76	8.38	6.41	7.81	0	6.93	5.48	6.55	5.18	8.56	8.88
18	9.56	10.38	11.08	10.7	10.93	10.15	10.73	10.83	11.16	10.8	11.08	3.33	4.61	1.9	1.3	2.63	6.95	0	3.45	2.11	2.71	2.25	5.98
19	10.78	8.93	9.63	9.25	9.48	8.68	9.26	9.38	9.7	9.33	9.63	2.28	6.76	4.91	2.51	3.21	5.5	3.45	0	4.26	1.61	4.7	8.13
20	9.11	10.05	10.75	10.36	10.6	9.81	10.4	10.5	10.83	10.46	10.75	3.28	2.75	3.61	1.85	3.73	6.63	2.08	4.38	0	2.95	3.71	4.13
21	9.6	8.63	9.33	8.96	9.18	8.4	8.98	9.1	9.41	9.05	9.35	1.21	5.43	4.21	2.01	2.7	5.21	2.75	1.63	2.93	0	4.18	6.8
22	11.28	12.1	12.8	12.41	12.65	11.86	12.45	12.55	12.88	12.51	12.8	5.33	6.23	2.33	3	3.26	8.66	2.26	4.78	3.73	4.25	0	7.61
23	7.63	9.38	10.08	10.03	10.25	12.26	9.03	9.2	9.38	9.16	8.78	7.4	1.65	7.71	5.96	7.83	8.98	6.2	8.48	4.38	7.05	7.81	0

, providing the foundational inputs for subsequent optimization calculations.

The minimum travel time observed was 0.8 h, corresponding to the delivery route from Ban None Champa to Ban Fang Deng in Lao PDR, whereas the maximum transport duration reached 12.88 h, encompassing the route from Kampot in Cambodia to Soukhoumma in Lao PDR. Discrepancies between outbound and return distances were occasionally noted, a phenomenon attributable to the route calculations implemented by the Google Maps platform, which are conditioned on prevailing traffic patterns and may not accurately reflect the actual trajectories traversed. Accordingly, the resulting distance matrix, generated from data acquired via the Google Maps application, provides not only a representation of contemporary traffic dynamics but also a methodological basis for the determination of optimal travel routes with enhanced precision and reliability.

To quantify transport efficiency, the transport time determinants (${lt}_{jk}$) from Table 7 were normalized by the corresponding transferable path parameters in Table 5, yielding a measure of transportability presented in

Table 8. Symmetrical Rate matrix (h/100 km) via road line. Source: the Authors.

(j, k)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23
1	0	1.65	1.68	1.71	1.68	1.85	1.6	1.68	2.21	1.66	1.63	1.7	1.52	1.48	1.66	1.77	1.72	1.64	1.72	1.6	1.71	1.66	1.61
2	1.69	0	1.79	2.07	1.75	2.32	1.88	2.38	2.39	2.2	1.9	1.6	1.55	1.48	1.65	1.74	1.56	1.63	1.61	1.6	1.63	1.65	1.62
3	1.7	1.79	0	1.91	1.77	2.12	1.8	1.95	2.2	1.89	1.82	1.61	1.56	1.5	1.66	1.74	1.6	1.64	1.62	1.61	1.64	1.65	1.63
4	1.73	2	1.88	0	1.78	2.22	1.98	2.15	2.3	2.08	1.94	1.61	1.57	1.49	1.65	1.74	1.59	1.64	1.61	1.6	1.63	1.65	1.64
5	1.69	1.73	1.76	1.69	0	2.03	1.76	1.89	2.14	1.84	1.78	1.59	1.56	1.48	1.64	1.73	1.56	1.62	1.6	1.59	1.62	1.64	1.62
6	1.89	2.36	2.15	2.24	2.08	0	2.24	2.35	3.03	2.3	2.17	1.65	1.68	1.51	1.69	1.78	1.68	1.67	1.65	1.63	1.68	1.68	1.73
7	1.64	1.83	1.82	1.98	1.8	2.22	0	3.26	2.25	2.76	1.86	1.61	1.53	1.49	1.65	1.74	1.58	1.63	1.61	1.6	1.63	1.65	1.61
8	1.72	2.28	1.95	2.13	1.92	2.31	3.67	0	2.36	5.33	2.28	1.62	1.55	1.5	1.67	1.76	1.62	1.65	1.63	1.62	1.65	1.66	1.63
9	2.22	2.42	2.22	2.31	2.18	3.06	2.3	2.39	0	2.35	2.22	1.92	1.69	1.66	1.9	1.97	2.29	1.86	1.87	1.83	1.91	1.85	1.76
10	1.7	2.3	1.96	2.12	1.91	2.3	3.4	5.33	2.35	0	2.12	1.62	1.55	1.5	1.67	1.76	1.61	1.65	1.63	1.61	1.65	1.66	1.62
11	1.7	1.85	1.81	1.93	1.8	2.14	1.81	2.11	2.2	2.08	00	1.62	1.54	1.5	1.66	1.75	1.6	1.64	1.62	1.61	1.64	1.66	1.61
12	1.69	1.62	1.63	1.63	1.61	1.66	1.63	1.65	1.93	1.64	1.64	0	1.56	1.43	2.09	2.14	1.66	1.92	1.84	1.9	1.91	1.91	1.61
13	1.54	1.56	1.58	1.58	1.58	1.69	1.55	1.56	1.7	1.56	1.55	1.62	0	1.34	1.56	1.68	1.69	1.55	1.71	1.46	1.58	1.57	2.03
14	1.48	1.5	1.51	1.5	1.5	1.52	1.5	1.52	1.67	1.51	1.51	1.43	1.31	0	1.3	1.42	1.46	1.17	1.43	1.33	1.43	2.4	1.38
15	1.65	1.66	1.67	1.66	1.65	1.7	1.66	1.68	1.9	1.67	1.67	2.06	1.48	1.27	0	2.56	1.7	2.39	2.09	1.91	2.22	1.9	1.55
16	1.77	1.75	1.75	1.75	1.74	1.79	1.75	1.77	1.98	1.77	1.76	2.12	1.71	1.43	2.59	0	1.84	2.07	2.18	2.21	2.28	1.95	1.73
17	1.7	1.59	1.62	1.61	1.58	1.68	1.61	1.65	2.29	1.63	1.63	1.64	1.69	1.45	1.7	1.84	0	1.67	1.64	1.62	1.68	1.68	1.75
18	1.64	1.64	1.65	1.65	1.64	1.68	1.65	1.66	1.86	1.66	1.65	1.9	1.5	1.14	2.39	2.03	1.67	0	1.87	1.9	2.03	1.82	1.56
19	1.71	1.62	1.63	1.63	1.62	1.65	1.63	1.64	1.87	1.63	1.64	1.8	1.64	1.42	2.05	2.15	1.64	1.86	0	1.98	1.99	1.8	1.67
20	1.46	1.62	1.63	1.62	1.61	1.65	1.62	1.64	1.84	1.63	1.63	1.91	1.36	1.32	1.96	2.02	1.64	1.87	2.03	0	1.84	1.76	1.49
21	1.71	1.64	1.65	1.65	1.64	1.69	1.65	1.67	1.92	1.66	1.66	1.91	1.52	1.43	2.19	2.28	1.69	2.05	2.01	1.84	0	1.81	1.58
22	1.66	1.67	1.67	1.67	1.66	1.7	1.67	1.68	1.86	1.68	1.68	1.94	1.53	2.3	1.97	1.98	1.7	1.83	1.83	1.77	1.84	0	1.58
23	1.6	1.6	1.62	1.63	1.62	1.72	1.6	1.61	1.73	1.6	1.6	1.67	2.09	1.41	1.62	1.71	1.73	1.61	1.74	1.57	1.63	1.62	0

. The analysis revealed substantial variation in transport rates, with the slowest being 5.33 h per 100 km from Ban None Champa to Ban Fang Deng in Lao PDR, and the fastest being 1.14 h per 100 km from Kampong Speu to Preah Sihanouk in Cambodia.

The optimal transferable path was encoded via a binary indicator where $y_{jk}$ = 1 for the fastest path and $y_{jk}$ = 0 otherwise. Following the Extended Traveling Salesman Problem framework based on distance and time or ETSPDT (Figure 3), fuel consumption along these paths was computed and is reported in

Table 9. Symmetrical Fuel consumption matrix (L) by using truck. Source: the Authors.

(j, k)	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23
1	0	20.96	27.46	26.9	28.91	35.17	17.44	19.21	123.06	18.98	15.25	79.01	56.39	96.31	82.94	101.53	51.73	87.21	99.72	83.16	88.47	102.5	68.02
2	21.06	0	6.45	27.24	7.92	58.35	3.32	17.9	76.59	17.12	6.73	66.91	72.64	102.33	89.11	102.93	31.34	93.58	80.05	89.43	77.64	108.67	84.16
3	27.33	6.45	0	12.44	14.38	91.56	9.69	10.67	109.74	10.24	13.01	73.46	78.99	108.99	95.62	109.39	37.92	100.1	86.6	95.94	84.16	115.18	90.46
4	26.76	27.06	12.27	0	6.91	73.62	9.18	45.03	103.62	44.21	12.52	70.14	78.83	105.54	92.19	105.98	34.52	96.67	83.16	92.66	80.73	111.79	90.1
5	28.67	7.82	14.28	6.5	0	87.35	11.05	12.08	117.98	11.63	14.39	71.67	80.31	106.92	93.82	107.69	36.01	98.28	84.69	94.19	82.34	113.38	91.67
6	35.33	58.62	91.79	73.76	87.78	0	74.48	75.97	91.33	75.19	91.09	65.71	100.82	101.56	87.8	101.29	30.2	92.33	78.84	88.26	76.38	107.38	111.6
7	17.57	3.23	9.78	9.18	11.35	74.27	0	4.56	84.34	4.28	3.32	70.09	69.02	105.46	92.19	105.96	34.47	96.63	83.11	92.61	80.73	111.98	80.51
8	19.4	17.77	10.67	44.95	12.25	75.77	4.62	0	85.89	1.46	21.3	71.24	71.05	107.05	93.51	107.17	35.63	97.75	84.49	93.89	81.84	113.08	82.43
9	123.13	76.64	109.96	103.69	118.37	91.44	84.55	86.09	0	85.32	101.18	76.04	74.71	114.8	98.04	110.37	167.17	102.75	89.35	99.26	86.49	118	85.32
10	19.08	17.27	10.57	44.37	12.07	75.19	4.4	1.46	85.32	0	20.93	71.05	70.92	106.85	93.29	106.98	35.39	97.56	84.06	93.58	81.61	112.85	82.19
11	15.59	6.55	13.01	12.43	14.58	90.85	3.23	21	100.96	20.84	0	73.39	67.12	109.14	95.66	109.41	37.77	99.91	86.4	96.03	83.97	115.25	78.44
12	78.66	68.11	74.64	71.33	73	66.5	71.4	72.75	76.46	72.22	74.92	0	51.26	40.76	113.96	157.25	37.44	30.97	21.51	29.79	11.15	48.4	64.25
13	57.12	73.09	79.94	79.45	81.38	101.4	69.74	71.63	75.01	71.11	67.57	54.04	0	50.16	40.48	58.69	71.3	42.66	64.85	25.52	50.57	57.42	67.02
14	96.37	104.07	110.78	107.21	109.02	102.65	107.27	108.83	115.59	108.15	110.92	41.07	47.81	0	21.41	34.95	73.02	12.19	42.02	28.89	35.69	90.1	62.15
15	82.38	89.91	96.41	92.8	94.86	88.12	92.99	94.29	97.87	93.85	96.48	113.54	37.37	20.47	0	67.72	58.8	48.32	106.13	16.59	80.24	26.74	50.48
16	101.24	103.45	109.92	106.41	108.44	101.5	106.69	107.88	110.34	107.4	109.93	156.07	57.38	35.25	68.11	0	72.11	111.99	130.6	148.61	104.35	29.53	70.2
17	51.2	31.97	38.52	35.13	36.9	30.31	35.32	36.56	167.17	35.99	38.73	36.87	71.08	71.99	58.8	72.11	0	63.34	49.84	59.36	47.39	78.38	81.81
18	86.97	94.51	101.02	97.48	99.45	92.89	97.79	98.89	103.02	98.58	101.06	30.71	40.46	11.03	48.32	111.56	63.57	0	31.84	19.46	115.01	20.77	53.55
19	98.98	81.01	87.49	83.99	85.9	79.18	84.06	85.42	89.53	84.84	87.58	21.04	61.56	41.41	105.7	130.29	50.07	31.85	0	39.14	14.77	43.37	74.34
20	78.45	91.07	97.61	93.93	95.96	89.38	94.36	95.49	100	95.07	97.65	30.23	22.23	28.18	17.01	158.94	60.3	19.2	185.91	0	27.24	34.19	36.1
21	88.13	78.57	85.08	81.69	83.51	76.94	81.88	83.2	86.74	82.64	85.38	11.15	48.05	35.69	80.23	104.35	47.72	116.04	69.79	27.05	0	38.59	61.14
22	103.03	110.6	117.09	113.45	115.51	108.77	113.86	114.93	118.91	114.53	117.12	49.07	55.21	89.44	27.57	29.94	79.47	20.86	44.13	34.39	39.24	0	68.43
23	68.89	84.79	91.34	91.06	92.9	112.7	81.49	83.3	86.29	82.82	79.31	67.61	68.59	64.7	54.08	71.92	82.59	56.16	78.04	39.28	64.02	70.76	0

. The results indicate a minimum fuel usage of 1.46 L for Ban None Champa to Ban Fang Deng in Lao PDR, while the maximum consumption reached 185.91 L from Kampong Chhnang to Svay Rieng in Cambodia, reflecting the combined influence of distance, travel time, and path characteristics. Furthermore, the operational functions of the ETSPDT were evaluated, with energy consumption optimized and quantified based on the transport time for each movement along the specified routes, as summarized in

Table 10. The path for transporting following ETSPDT. Source: the Authors.

Order	j to k	Distance (km): ${\mathit{y}}_{\mathit{j}\mathit{k}}{\mathit{d}}_{\mathit{j}\mathit{k}}$	Time (h)	Velocity (km/h): ${\mathit{v}}_{\mathit{j}\mathit{k}}$	Fuel Consumption Rate (L/km): ${\mathit{e}}_{\mathit{j}\mathit{k}}$ Stage 1	Fuel Consumption Rate (L/km): ${\mathit{e}}_{\mathit{j}\mathit{k}}$ Stage 2	Amount (L): ${\mathit{e}}_{\mathit{j}\mathit{k}}{\mathit{y}}_{\mathit{j}\mathit{k}}{\mathit{d}}_{\mathit{j}\mathit{k}}$
1	1 to 11	103	1.7	61		0.15	15.3
2	11 to 7	19	0.4	55		0.17	3.2
3	7 to 2	19	0.4	55		0.17	3.2
4	2 to 3	39	0.7	56		0.17	6.5
5	3 to 10	59	1.1	53		0.17	10.2
6	10 to 8	2	0.1	19	0.98		1.5
7	8 to 5	69	1.3	52		0.18	12.3
8	5 to 4	42	0.7	59		0.15	6.5
9	4 to 17	240	3.8	63		0.14	34.5
10	17 to 6	196	3.3	59		0.15	30.3
11	6 to 12	436	7.2	60		0.15	65.7
12	12 to 21	63	1.2	52		0.18	11.2
13	21 to 20	159	2.9	54		0.17	27.1
14	20 to 15	94	1.9	51		0.18	17.0
15	15 to 14	215	2.8	78		0.10	20.5
16	14 to 18	166	2.0	85		0.07	12.2
17	18 to 22	123	2.3	55		0.17	20.8
18	22 to 16	164	3.3	50		0.18	29.9
19	16 to 13	365	6.3	58		0.16	57.4
20	13 to 19	411	7.1	58		0.16	64.9
21	19 to 23	485	8.1	60		0.15	74.3
22	23 to 9	540	9.4	58		0.16	86.3
23	9 to 1	140	3.1	45	0.88		123.1
	Total	4148.7	70.8				733.8

. The transport route can be represented as the ordered sequence:

As exhibited in Table 10, this representation highlights the sequential cross-border movements, clearly distinguishing the segments within Thailand, the Lao PDR, and Cambodia, while preserving all intermediate locations for precise operational analysis.

Route: Ubon Ratchathani (TH) → Phonthong, Pakse, Sanasomboun, Ban None Champa, Ban Fang Deng, Paksong, Batiengchaleunsouk (Lao PDR) → Stung Treng, Pathouphone (Lao PDR), Tboung Khmum, Prey Veng, Kampong Chhnang, Phnom Penh, Preah Sihanouk, Kampong Speu, Kampot, Kandal, Battambang, Svay Rieng, Pailin, Soukhoumma} (Cambodia)→Ubon Ratchathani (TH).

Additionally, the whole transport distance was 4148.7 km while the gross transportation time was 70.8 h; this resulted in the vehicle movability being 1.7 h per 100 km, which was able to be transformed to a whole vehicle speed for this trip of 58.6 km per hour. Subsequently, the fuel consumption was its cost effectiveness function, which in total was predicted to be 733.82 L regarding carrying by truck in order to completely deliver goods to all destinations following the ETSPDT concept.

4.4. Effective Comparison

This section presents a systematic comparative analysis of the classical Traveling Salesman Problem, defined solely by distance (TTSPD), and its extended formulation, which incorporates both distance and temporal dimensions (ETSPDT). Positioned within the context of international delivery operations, this comparison not only advances the central objectives of the study but also highlights the operational significance of integrating time-sensitive constraints into route optimization. From the perspective of symmetry, the analysis further reveals how structural regularities and balanced decision patterns can be leveraged to improve the design of efficient distribution strategies. In supply chain and logistics management, such symmetry-based considerations are particularly valuable, as they contribute to more coherent planning, reduced operational complexity, and improved robustness in network coordination. Building on this foundation, the study identifies the most effective alternative managerial strategy for international distribution, thereby providing actionable insights that may enhance decision-making efficiency, strategic flexibility, and overall performance within complex logistical systems.

As shown in

Table 11. The comparison of effectiveness between TTSPD and ETSPDT. Source: the Authors.

Description	TTSPD	ETSPDT	Different Evaluation
			ETSPDT-TTSPD	Rate (%)
1. Overall transport distance (km)	2850.6	4148.7	1298.1	45.5
2. Overall transport duration (h)	52.7	70.8	18.1	34.3
3. Average vehicle speed (km/h)	54.1	58.6	4.5	8.3
4. Average fuel consumption (L/km)	0.42	0.18	−0.24	−57.1
5. Total energy consumption (fuel in liters)	1193.5	733.8	−459.7	−38.5

, a comparative analysis is performed between the traditional traveling salesman problem based solely on distance (TTSPD) and its extended formulation incorporating both distance and time (ETSPDT). Both models are applied to an international transportation problem with a uniform set of delivery destinations and subsequently evaluated within their respective mathematical frameworks. The ETSPDT formulation explicitly accounts for speed variability along routes as a function of travel time and distance, thereby targeting fuel consumption minimization. In contrast, TTSPD is restricted to shortest-path optimization. Although both approaches aim to reduce transportation costs via fuel usage, their operational mechanisms differ substantially.

Quantitatively, ETSPDT results in increases in total distance and travel time of approximately 45.5% and 34.3%, respectively, yet achieves a higher average velocity (+8.3%). Crucially, this leads to a markedly lower average fuel consumption rate (−57.1%) and a corresponding reduction in fuel costs (−38.5%) relative to TTSPD. These results indicate that ETSPDT delivers superior energy efficiency and constitutes a more effective framework for minimizing transportation costs.

5. Discussion

Within the domain of international land logistics services, this study develops a set of mathematical models designed to capture the structural complexities of cross-border delivery operations, with particular attention to cost-effectiveness assessment. In this framework, vehicle energy consumption, especially in truck transportation, is employed as a practical proxy for estimating overland logistics costs across interconnected networks. From a symmetry perspective, the models reflect balanced and structured relationships among routing, allocation, and energy-use parameters, thereby revealing underlying regularities in logistics systems that may otherwise remain obscured. Such symmetry-based formulations are particularly valuable in supply chain management, as they facilitate the identification of optimal operational patterns while preserving consistency across multiple international destinations. Among the proposed alternatives, the model demonstrating the greatest cost efficiency is ultimately selected as a decision-support tool for managerial planning, supporting strategic vehicle allocation and routing decisions in complex cross-border logistics environments.

To ensure practical relevance, the study begins with an examination of critical challenges within Thailand’s contemporary economic landscape. This is followed by a comprehensive review of the extant literature to identify significant research gaps in the field of international road freight transportation. Subsequently, a series of scientific models is developed to address the identified challenges in international land logistics. Both conventional and newly proposed models are systematically represented through flowcharts to facilitate algorithmic implementation, prior to the collection of secondary data from Google Maps for numerical experimentation. All formulated mathematical models are then rigorously analyzed and comparatively evaluated in terms of their cost-effectiveness. Finally, the study articulates several key implications for both academic research and practical application, thereby providing a robust foundation for the future advancement of planning and operational strategies in international land transportation services.

5.1. Research Implications

To address the development of an alternative model for optimizing international land logistics via an extended Traveling Salesman Problem (TSP), this study builds upon the classical TSP framework and augments it with additional constraints to enable energy consumption estimation. Two models are formulated. The first is an optimization model that minimizes total travel distance: subject to standard TSP constraints. The second extends this formulation by incorporating travel time and fuel consumption relating distance and travel time between nodes which results in fuel consumption. Empirical data are derived from administrative records in Thailand and the Google Maps platform, from which pairwise distances and travel times are extracted. The resulting dataset is analyzed using quantitative and visual methods. From the perspective of symmetry, the proposed modeling approach seeks to preserve structural consistency across routing decisions, cost relationships, and network configurations. In logistics systems, symmetry provides a useful analytical lens through which recurring patterns and balanced interdependencies can be identified, particularly in multi-node supply chain networks.

Results demonstrate that the distance-based model (TTSPD) yields higher energy consumption than the extended model (ETSPDT). In TTSPD, routing decisions depend solely on distance minimization, implicitly determining vehicle speed. In contrast, ETSPDT jointly considers distance and time, enabling the explicit optimization of fuel consumption. This highlights a key limitation of single-criterion models: minimizing distance alone does not ensure energy efficiency. In practice, transportation systems are influenced by dynamic factors such as traffic conditions, weather, and infrastructure quality, all of which affect travel time. Consequently, optimal real-world routes often diverge from shortest-path solutions. Incorporating time as a decision variable allows the model to better approximate realistic operating conditions and improve energy efficiency.

Furthermore, the joint variation of distance and time significantly affects route selection prior to final path construction. These findings support the hypothesis that distance, time, and fuel consumption jointly define the constraint structure governing transport efficiency. The results validate the multi-criteria framework illustrated in Figure 1, demonstrating that integrated optimization is more cost-effective than single-objective approaches. Overall, this study contributes to the literature on international land logistics by proposing a tractable multi-criteria TSP extension with practical relevance for transport planning, particularly in developing regions.

5.2. Practical Implications

As illustrated in Table 11, the effectiveness of the Traveling Salesman Problem (TSP) in addressing challenges associated with international land logistics is clearly demonstrated. In practice, planners in this domain are required not only to optimize vehicle routing in order to minimize travel distances for cross-border deliveries, but also to conduct comprehensive evaluations that integrate multiple interrelated factors, including travel distance, transit time, and fuel consumption rates, all of which dynamically influence overall vehicular energy expenditure. Moreover, in the context of international road networks, logistics planners may adopt the Extended Traveling Salesman Problem based on the Distance and Time (ETSPDT) model, as it has been shown to improve transportation cost-efficiency while accommodating the added complexity of temporal and spatial constraints.

5.3. Limitation and Future Direction

This study is not without limitations. First, the secondary data employed in the analysis were randomly collected from the Google Maps application, with a focus on predominantly developing countries in the Southeast Asian region. As a result, the characteristics and performance of road networks in developed countries may produce substantially different findings. Second, the scope of the investigation was restricted to international road freight transport conducted by trucks. This limitation reflects a fundamental mathematical constraint of the proposed models: although they may generate an “optimal” route based on a single snapshot of real-time distance and travel-time data obtained from Google Maps, that route can quickly become suboptimal or even infeasible as traffic conditions change. Accordingly, cross-border land transportation in other contexts may involve distinct route optimization strategies and exhibit different levels of operational efficiency. Taken together, these limitations highlight the need for future research to further advance and refine the existing body of knowledge in international land logistics.

6. Conclusions

This study seeks to develop a novel mathematical model for addressing persistent challenges in international land logistics by evaluating its cost efficiency relative to conventional routing approaches. The analysis is formulated as an extension of the Traveling Salesman Problem (TSP) and applied in the context of truck-based freight transportation. To support the model, supplementary transport data were randomly extracted from administrative records in Thailand and augmented with spatial and temporal logistics variables, namely distance and travel time, obtained from the Google Maps platform.

Traditionally, international land logistics and land-based delivery systems have prioritized the minimization of total travel distance under the assumption that shorter routes naturally reduce travel time, fuel consumption, and overall transportation cost. However, in real-world road networks, vehicle movement is rarely governed by distance alone. Traffic congestion, road surface conditions, weather variability, and other stochastic disruptions significantly alter travel time and energy expenditure. Although drivers often attempt to select the shortest routes, they may ultimately experience longer delays and greater fuel use due to congestion and network instability. In response, alternative routes such as bypasses are frequently adopted to improve efficiency and avoid bottlenecks.

This gap between theoretical routing assumptions and practical transportation behavior motivates the present study to propose an extended mathematical framework based on the TSP. The proposed model jointly optimizes travel distance and travel time while placing primary emphasis on minimizing vehicle fuel consumption. These objectives are formulated as complementary optimization criteria within the broader setting of international land logistics services. In this respect, the model reflects an important structural principle in logistics optimization: the balance between symmetry and asymmetry in transport networks. While classical routing problems often assume symmetrical distance relationships between locations, actual supply chain systems are frequently asymmetric due to direction-dependent travel times, road conditions, border procedures, and congestion patterns. Recognizing and modeling such asymmetry is essential for producing solutions that are both theoretically sound and operationally realistic.

To develop the framework, a comprehensive review of the relevant literature was first conducted. Thereafter, the traditional Traveling Salesman Problem based solely on distance (TTSPD) was reformulated in accordance with its classical structure. In parallel, an extended model incorporating both distance and time (ETSPDT) was developed. The two models were then tested and evaluated through computational analysis using the Solver Add-in in Microsoft Excel 2024, following their respective algorithmic flow structures. Secondary data, including delivery locations, road network linkages across ASEAN countries, and corresponding travel distances and times derived from Google Maps, were randomly collected and integrated into the analysis. Besides, the results demonstrate the effectiveness of both TTSPD and ETSPDT in representing vehicle energy consumption, and the key findings may be summarized as follows.

Firstly, the study establishes that optimizing transportation in international land logistics should not rely solely on minimizing travel distance as a proxy for reducing fuel consumption. Instead, it requires an integrated consideration of both travel distance and travel time in relation to vehicle speed, as variations in velocity significantly influence energy consumption rates. Consequently, distribution planning based exclusively on distance is insufficient for achieving optimal energy efficiency in practical logistics operations.

Secondly, the findings indicate that optimizing vehicle energy consumption can be more effectively achieved through an extended TSP framework that incorporates both distance and time. This approach provides a robust alternative model for logistics planners managing international product distribution. Notably, the proposed model demonstrates substantial potential for reducing vehicle energy costs, particularly in the context of cross-border road transportation planning.

From a broader supply chain management perspective, the study contributes to the growing body of literature on routing efficiency, operational resilience, and sustainability. The concept of symmetry is particularly relevant here, as it provides a useful analytical lens for examining whether transport networks behave uniformly in both directions. Besides this, incorporating symmetry considerations into routing models enhances both theoretical rigor and managerial relevance. As a result, combining symmetry-aware modeling with time–distance optimization allows researchers to develop more accurate and actionable decision-support tools for modern logistics systems.