Circuit Analysis Approach for Sustainable Routing Optimization with Multiple Delivery Points

Abstract

This paper introduces a novel methodology for vehicle routing services called Route Optimization with Multiple Delivery Points (ROMP), which works by modeling urban street networks as analog electrical circuits. This methodology translates road networks into a linear electrical circuit where the resistances of circuit branches represent parameters like vehicular flow and street length, derived from geographic positions between intersections. By applying Modified Nodal Analysis (MNA) to this circuit, ROMP identifies high-current paths that closely approximate minimal travel distances. The practical performance of ROMP is demonstrated through three case studies, showing its potential to yield shorter routes and faster route-finding compared to OpenRouteService (ORS). The resultant improvements can lead to fuel savings, reduced labor costs, and enhanced logistics operations, particularly in applications involving a single origin and multiple delivery points, such as goods delivery and patient transport. In addition, this proposal supports sustainability by optimizing routes, which helps reduce the environmental impact of transportation and lower greenhouse gas emissions. Furthermore, shorter travel distances and improved efficiency promote better energy use, enhancing air quality and urban sustainability. Future work aims to integrate new street models and real-time traffic data to expand ROMP’s applicability in vehicle routing research.

1. Introduction

The United Nations defines sustainable development as growth that meets current needs without compromising those of future generations. To achieve this goal, it is essential to balance economic growth, social inclusion, and environmental protection. Governments play a critical role in ensuring energy security, supporting initiatives to mitigate climate change, and improving air quality in densely populated areas. In fact, the transportation sector significantly influences these aspects [1,2]. A key challenge for transportation systems is finding environmentally friendly solutions, as high pollution levels in cities have been linked to serious health problems, including cardiorespiratory diseases, cancer, and increased mortality [3,4]. Moreover, economic growth has contributed to the rise in polluting gas emissions, with serious consequences for health and the environment. For example, prolonged exposure to fine particulate matter (PM_2.5μm) and ozone (O₃) has been responsible for millions of annual deaths and new cases of asthma in children [5,6,7]. Transportation, particularly in urban areas, significantly contributes to air pollution. Economic development has increased the demand for transportation, leading to more private and commercial vehicle use and higher emissions of polluting gases, including carbon dioxide (CO₂), a primary greenhouse gas responsible for climate change [8]. According to the European Commission, private and commercial vehicles generate approximately 15% of CO₂ emissions in the European Union [8]. Air pollution is a significant risk factor for cardiovascular diseases [9]. Although less attention has been given to environmental noise, which often coexists with air pollution in urban areas, the World Health Organization (WHO) stated in their 2018 Environmental Noise Guidelines for the European Region that traffic noise increases the risk of heart disease. In many cases, a large part of the population is exposed to traffic noise levels that exceed recommended thresholds [10]. Vehicular transportation noise is an increasingly recognized environmental pollutant. This noise coexists with air pollution in urban settings, reflecting vehicular traffic as a major source of both types of exposure [11]. According to data from the Global Burden of Disease (GBD) study [12], the WHO [13], and the Global Health Observatory (GHO) [14], the main causes of diseases have substantially shifted from infectious to non-communicable diseases over the last three decades, with cardiovascular diseases caused by atherosclerosis or metabolic diseases being the most important category [11]. The Lancet Commission on Pollution and Health concluded that “environmental pollution is the most important cause of diseases and premature deaths in the world” and estimated that in 2015 alone, air pollution caused 9 million deaths. New evidence shows that even low levels of PM_2.5μm air pollution can increase the risk of death [11]. Although scientific and medical efforts in the past have focused on traditional cardiovascular risk factors such as diabetes mellitus and smoking, the Global Burden of Disease study suggests that environmental factors play an important role in the development of chronic non-communicable diseases and, therefore, contribute substantially to global mortality [11]. The global transportation system heavily relies on fossil fuels like gasoline, diesel, petroleum, and natural gas, which are major sources of CO₂ emissions. In 2016, fossil fuels accounted for 91% of the total energy consumed in the U.S. transportation sector. Although alternative energy vehicles, such as electric, hydrogen, and solar vehicles, are promising options for reducing CO₂ emissions in transportation, their widespread adoption is hindered by technical and economic challenges, such as high investment costs, short travel ranges, and a lack of recharging stations [15]. The rise in transportation has led to alarming levels of pollution globally, adversely affecting both the environment and human health. Researchers are developing solutions to limit fuel consumption in vehicles to reduce pollutant emissions. In [16], a formulation of the Pollution-Routing Problem (PRP) aims to minimize fuel consumption and travel distance. A new solution based on the Non-Dominated Sorting Genetic Algorithm-II (NSGA-II) is proposed, as the problem requires simultaneous optimization of multiple routes. In [17], Multi-Destination Vehicle Route Planning (MDVRP) is proposed to optimize travel time and search for on-street parking spaces, regardless of the destination order. MDVRP uses the Time-Dependent Traveling Salesman Problem (TDTSP) [18] and Free Parking Assignment (TDTSP-FPA) [17] to find the fastest routes between destinations and assign free parking spaces, aiming to minimize total travel time for drivers. Proposals such as carpooling or ridesharing have also been reported [19], where a driver and one or more passengers share part of their trips using the same vehicle. Carpooling typically involves starting from an origin or pickup point and reaching a destination or drop-off point, with optional stops along the way. Although carpooling offers several economic and ecological advantages, its integration into transportation travel planners has not been studied in detail [19]. Researchers currently use genetic algorithms, optimization techniques based on natural evolution, to solve the route optimization problem [20]. In the context of route optimization, a genetic algorithm could be used to find the best possible route for a set of given destinations. The algorithm begins by generating an initial population of possible solutions, represented as strings of genes [20]. These solutions are then evaluated based on their quality, such as total traveling distance and time, or other relevant metrics. There have also been proposals to improve transportation route planning for food storage and distribution to various service centers, where products are offered to customers in specific regions [20]. The transportation cost is estimated for each potential delivery route to determine the most resource-efficient option. Solutions to such food distribution logistics problems are particularly important to ensure efficient and timely food delivery to places where it is needed, especially in regions where distribution is a challenge due to geographic, climatic, or socioeconomic factors. Route optimization studies have also been carried out by transportation companies using the savings matrix method, based on data collection such as traveling distance and route design [21]. In this context, the use of multi-connection resistive grids is proposed in [22] to obtain delivery routes. However, the analysis does not consider streets with unidirectional vehicular flow, which is particularly relevant, as road networks consist of streets allowing vehicular flow in both directions. Including the permitted directions from the street network map allows for more realistic routes to be calculated. Considering the environmental impact is crucial when planning and conducting travel on road networks. Travel time directly impacts the amount of pollutants emitted into the atmosphere. Therefore, it is important to implement methodologies to find the best routes within road networks, reducing total transportation distance, time and polluting gas emissions when traveling to different destinations. This is particularly important for the delivery of essential supplies such as drinking water, food, and medicine to warehouses, distribution centers, hospitals, and other key resource distribution points [23]. Reducing polluting gas emissions in the transportation sector is crucial to mitigate the effects of climate change and protect human and environmental health. The document is structured as follows. Section 2 introduces the ROMP methodology. Section 3 illustrates the proposed methodology with an example. Section 4 presents three case studies corresponding to different cities. Section 5 provides a detailed discussion and comparison in terms of distance and route-finding time between ROMP and the ORS library. Finally, Section 6 presents the conclusions.

2. Optimizing Delivery Routes from a Single Origin to Multiple Destinations

The phenomenon of current flowing more intensely through the path of least resistance in a circuit is analogous to route planning in road networks, where the shortest route is prioritized [24,25]. Various studies have explored modeling vehicle movement using electrical circuit principles. In [26], a potentiometer-based street model is used to adjust current magnitude to represent traffic distribution. Similarly, ref. [27] equates voltages and currents in a circuit to travel time and traffic intensity, defining resistance as the ratio between them. The study also considers multiple starting points converging at a single destination, though only as a theoretical resistive network without cartographic data. A probabilistic Markov model has been used to estimate traffic flow [28], while ref. [29] assumes a single central node where all vehicles converge. Other models, such as [30,31], use electrical circuits to simulate steady-state and transient traffic conditions. However, these approaches focus on reducing travel time under specific conditions using simplified, hypothetical maps, considering only one-way streets and lacking geographic or cartographic details of real urban road networks.

In this work, the proposed ROMP is a methodology for calculating transportation routes from a single origin to multiple destinations within a road network. The origin and destinations correspond to the intersection points between streets, which are the nodes of the equivalent electrical circuit obtained from the map under analysis. The blocks form the branches of the circuit, and the resistive value R assigned to each branch is calculated based on the length of the street segment between two intersections using the following:

$$R=\rho {\displaystyle \frac{L}{A}},$$

(1)

The values for $\rho$ and A in this proposed work are $1 \mathrm{m}\mathrm{\Omega }$, and ${1 \mathrm{m}}^2$, respectively. L is the length of the block and is obtained as follows:

$$L=\sum _{i=1}^{n-1}{d}_{P_i{P}_{i+1}},$$

(2)

where $d_{P_i{P}_{i+1}}$ is the distance between two geographic points expressed in radians, calculated using the Haversine Formula (3) [32]. The difference in latitudes and longitudes that form the block are represented by (4) and (5). r represents the radius of the Earth, approximately 6371 km [33].

$$d_{P_i{P}_{i+1}}=2r{sin}^{-1}\left(\sqrt{si{n}^2\left({\displaystyle \frac{\Delta lat}{2}}\right)+cos\left(latitud{e}_{P_{i+1}}\right)cos\left(latitud{e}_{P_i}\right)si{n}^2\left({\displaystyle \frac{\Delta long}{2}}\right)}\right)$$

(3)

where,

$$\Delta lat=latitud{e}_2-latitud{e}_1$$

(4)

$$\Delta long=longitud{e}_2-longitud{e}_1$$

(5)

According to Ohm’s law, there is a relationship between the length of each block and its corresponding branch in the circuit, determined by the current flowing through them. In this context, the greater the length of the block, the lower the current flowing through its equivalent branch. This methodology allows a near-optimal route to be calculated by selecting, node by node, the branches with the highest current values.

For this methodology, the nodes of interest in the electrical circuit corresponding to the road network under analysis are selected, and the distances $d_{i,j}$ from each node to the rest of the selected nodes are calculated using EMBRP [34]. Consider the resistive network shown in Figure 1, where nodes $n_1$, $n_2$, $n_3$, $n_4$, $n_5$, and $n_6$ represent street intersections within the road network, and $R_1$, $R_2$, $R_3$, $R_4$, $R_5$, $R_6$, and $R_7$ represent the distances between these intersections. The nodes of interest in the resistive network in Figure 1 are $n_1$, $n_3$, and $n_5$. The next step is to calculate the distance from $d_{1,3}$. For this, the voltage source $V_E$ is placed at $n_1$ and $V_G$ at $n_3$, as shown in Figure 2. Then, using the stamps for conductance $G_i={\displaystyle \frac{1}{R_i}}$ and the voltage source shown in

Table 1. Conductance stamp.

	${\mathit{V}}_{\mathit{i}}$	${\mathit{V}}_{\mathit{j}}$	RHS
Node $\mathit{i}$	$G_{ij}$	$-G_{ij}$	0
Node $\mathit{j}$	$-G_{ij}$	$G_{ij}$	0

and

Table 2. Voltage source stamp.

	${\mathit{V}}_{\mathit{i}}$	${\mathit{V}}_{\mathit{j}}$	${\mathit{i}}_{\mathit{i}\mathit{j}}$	RHS
Node $\mathit{i}$	0	0	1	0
Node $\mathit{j}$	0	0	−1	0
Branch $\mathit{i}\mathit{j}$	1	−1	0	$E_{ij}$

, the Modified Nodal Analysis (MNA) is formulated.

The MNA system obtained for the electrical circuit in Figure 2 is given by the following:

$$\begin{array}{cccccccccc}[& \begin{array}{cccccccc}{G}_1+G_5& -G_1& 0& 0& 0& -G_5& 1& 0\\ -G_1& G_1+G_2+G_4& -G_2& 0& -G_4& 0& 0& 0\\ 0& -G_2& G_2+G_3& -G_3& 0& 0& 0& 1\\ 0& 0& -G_3& G_3+G_6& -G_6& 0& 0& 0\\ 0& -G_4& 0& -G_6& G_4+G_6+G_7& -G_7& 0& 0\\ -G_5& 0& 0& 0& -G_7& G_5+G_7& 0& 0\\ 1& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0& 0\end{array}& ]& [& \begin{array}{c}{V}_1\\ V_2\\ V_3\\ V_4\\ V_5\\ V_6\\ i_{VE}\\ i_{VG}\end{array}& ]& & [& \begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ V_E\\ V_G\end{array}& ]\end{array}$$

From the MNA system, the equation $Ax=b$ is formulated. By solving this equation or calculating the DC operating point, nodal voltages are obtained. Subsequently, branch currents are calculated. Finally, the Modified Current Comparison Algorithm (MLCCA) is used to obtain the route by selecting the branches with the highest current between $n_1$ and $n_3$, considering the allowed directions within the road network.

To calculate the distance $d_{1,5}$, the matrix A is not reformulated. Only the values corresponding to the contributions of the voltage sources $V_E$ and $V_G$ placed at $n_1$ and $n_5$ are updated. The rest of the matrix values and the vectors x and b are reused. Therefore, the updated MNA system is given by the following:

$$\begin{array}{cccccccccc}[& \begin{array}{cccccccc}{G}_1+G_5& -G_1& 0& 0& 0& -G_5& 1& 0\\ -G_1& G_1+G_2+G_4& -G_2& 0& -G_4& 0& 0& 0\\ 0& -G_2& G_2+G_3& -G_3& 0& 0& 0& 0\\ 0& 0& -G_3& G_3+G_6& -G_6& 0& 0& 0\\ 0& -G_4& 0& -G_6& G_4+G_6+G_7& -G_7& 0& 1\\ -G_5& 0& 0& 0& -G_7& G_5+G_7& 0& 0\\ 1& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0\end{array}& ]& [& \begin{array}{c}{V}_1\\ V_2\\ V_3\\ V_4\\ V_5\\ V_6\\ i_{VE}\\ i_{VG}\end{array}& ]& & [& \begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ V_E\\ V_G\end{array}& ]\end{array}$$

By solving $Ax=b$, the nodal voltages, branch currents, and the path between $n_1$ and $n_5$ are calculated. The process is repeated to calculate the distances $d_{3,1}$, $d_{3,5}$, $d_{5,1}$, and $d_{5,3}$. After calculating all the distances between the nodes, $n_1$ is chosen as the origin, and $n_3$ and $n_5$ as the delivery points or destinations. Routes are obtained by permutations $\mathcal{P}\left(n\right)$ of the delivery points through the following:

$$\mathcal{P}\left(n\right)=(n-1)!$$

(6)

$\mathcal{P}\left(n\right)$ refers to permutations of n elements with node $n_1$ fixed, in this case routes are follows;

$$[n_1,n_3,n_5,d_{1,3}+d_{3,5}],$$

$$[n_1,n_5,n_3,d_{1,5}+d_{5,3}],$$

where $d_{1,3}$ + $d_{3,5}$ and $d_{1,5}$ + $d_{5,3}$ represent the sum of the distances of each sub-route that form the route, giving priority to the shortest distance.

Figure 3 shows the block diagram for ROMP. The proposed methodology is composed of five main blocks: (a) selecting the OSM area, (b) calculating distances with EMBRP [34], (c) permutation and routing (d) obtaining the route, and (e) GeoJSON visualizer routing.

First, the area of interest is selected in OpenStreetMap (OSM) to obtain the geographic position data of the street network, which is used to create the equivalent linear electric circuit. The nodes corresponding to the geographic positions of interest are chosen in the circuit. Next, the distances $d_{i,j}$ between the nodes are calculated using EMBRP, where i and j represent the origin and destination of the corresponding sub-route.

After obtaining all the distances between the geographical positions, one is chosen as the origin, and the rest are delivery points. Routes are determined by permutations between delivery points. The total distance of each route is calculated as the sum of the individual distances of each sub-route. Then, prioritizing the route nearest to the shortest distance, the sub-routes that form the route are selected according to the order in which each delivery point should be visited, starting from the selected origin. Finally, the route is displayed in the GeoJSON viewer.

Although it is possible to obtain a route near the shortest distance between delivery points, other routes with longer distances visiting the same delivery points in a different order can also be found. Additionally, for the nodes of interest, it is possible to change the origin and the delivery points without the need to repeat the simulation, as it is not necessary to recalculate the distances between the nodes, representing a significant saving of computational resources.

The pseudocode Algorithms 1–3 correspond directly to the process illustrated in Figure 3. Algorithm 1 encapsulates the entire ROMP methodology, beginning with selecting an area in OpenStreetMap (OSM), followed by calculating distances between nodes of interest using the EMBRP method, generating route permutations, and finally selecting and visualizing the shortest route. Algorithm 2 specifically details the “Permutation and Routing” block from Figure 3, generating all possible permutations of the selected delivery points and calculating their respective total distances. Algorithm 3 corresponds to the “Get Route” block, selecting the optimal route by comparing distances and choosing the shortest route among the available permutations. Thus, these algorithms implement the conceptual schema presented in Figure 3, systematically achieving the shortest route for multiple delivery points.

In the following section, an illustrative example of the proposed methodology is presented.

Algorithm 1 ROMP routing calculation for Figure 3

1: Start  2: Select area in OSM  3: Obtain the data of the geographic positions of the streets.  4: Select nodes of interest  5: $nodes\leftarrow$ select_nodes_interest(N)  6: Calculate distances $d_{ij}$ from each node to the other $n-1$ nodes  7: $distances\leftarrow$ calculating_distances(nodes)  8: Generate possible routes by permutation and obtain their distances  9: $Routes\_possibles\leftarrow$ generate_permutations(nodes) 10: $distances\_routes\leftarrow$ calculate_distances_routes(routes_possibles, distances) 11: Obtain the route with the minimum distance 12: $route\leftarrow$ obtain_route(minimum_distance_route) 13: Display route 14: Display_route(Route with the minimum distance obtained) 15: End

Algorithm 2 Permutation and Routing

1: Entry: Set of nodes $N=\{n_1,n_2,\dots ,n_n\}$  2: Output: List of possible routes and their corresponding distances  3: Initialize list of routes and distances: $routes\_and\_distances\leftarrow \left[\right]$  4: Generate all permutations of the nodes: $routes\leftarrow$ generate_permutations(N)  5: for each route in routes do  6:       Calculate the total distance of the route:  7:       $distance\leftarrow$ Calculate_distance_route(route)  8:       Save the route together with its distance in the list:  9:       $routes\_and\_distances.push\left(\right[route,distance\left]\right)$ 10: end for 11: Return $routes\_and\_distances$

Algorithm 3 Get route

1: Input: List of routes and their distances $routes\_y\_distances$ 2: Output: Route with the shortest distance 3: $(route,distancia\_minima)\leftarrow rutas\_and\_distances\left[0\right]$ 4: for each $(route,distance)$ in $routes\_and\_distances$ do 5:       if $distance<distance\_minimum$ then 6:           $(route,distance\_minimum)\leftarrow (roure,distance)$ 7:       end if 8: end for 9: Return $(route,distancia\_minima)$

3. ROMP Illustrative Example

The proposed ROMP methodology is described through an illustrative example. The selected area of interest is shown by the dashed-green-line frame in Figure 4 and corresponds to the street network of a section of the map of the city of Xalapa in the state of Veracruz, Mexico, obtained from OpenStreetMap [35,36]. The node origin is depicted by a red icon, while the four delivery points are depicted by green icons. The latitude and longitude geographical positions of the origin and delivery points are available from OSM. The equivalent linear electric circuit obtained from the road networks for Figure 4 is shown in Figure 5. Origin $n_1$ corresponds to the geographical position marked with the red icon, while nodes $n_2$, $n_3$, $n_6$, and $n_{11}$ are delivery points corresponding to geographical positions marked with green icons. The resistive value of each branch that forms the circuit is shown in

Table 3. Resistive values for the circuit in Figure 5.

Resistance	Value ($\mathbf{\Omega }$)	Resistance	Value ($\mathbf{\Omega }$)	Resistance	Value ($\mathbf{\Omega }$)
$R_1$	96.845	$R_7$	101.072	$R_{13}$	109.467
$R_2$	95.873	$R_8$	94.289	$R_{14}$	103.586
$R_3$	94.628	$R_9$	96.651	$R_{15}$	90.012
$R_4$	98.023	$R_{10}$	94.176	$R_{16}$	77.815
$R_5$	159.817	$R_{11}$	95.871	$R_{17}$	81.318
$R_6$	102.322	$R_{12}$	154.728	$R_{18}$	155.762

.

After calculating the resistance values for each branch of the circuit, the voltage source $V_E=50$ volts is placed at $n_1$ and $V_G=0$ volts at node $n_2$. Subsequently, using the conductance and voltage source stamps as shown in Table 1 and Table 2, the Modified Nodal Analysis (MNA) and its corresponding system of Equations $Ax=b$ are obtained. For the electrical circuit in Figure 5, the system is given by the following:

$$\begin{array}{cc}[& \begin{array}{ccccccccc}0.0195& 0& 0& 0& 0& 0& 0& 0& -0.0104\\ 0& 0.02262& -0.01032& 0& 0& 0& 0& 0& 0\\ 0& 0& 0.0226& -0.0103& 0& 0& 0& 0& 0\\ 0& 0& -0.0103& 0.0270& -0.0104& 0& 0& -0.0062& 0\\ 0& -0.0064& 0& -0.0104& 0.0274& -0.0105& 0& 0& 0\\ 0& 0& 0& 0& -0.0105& 0.0170& 0& 0& -0.0064\\ 0& 0& 0& 0& 0& 0& 0.0309& -0.0102& 0\\ 0& -0.0098& 0& -0.0062& 0& 0& -0.0102& 0.0369& 0\\ -0.0104& -0.0097& 0& 0& 0& -0.0064& 0& 0& 0.0266\\ 0& 0& 0& 0& 0& 0& 0& -0.0106& 0\\ 0& 0& 0& 0& 0& 0& -0.0111& 0& 0\\ -0.0091& -0.0106& 0& 0& 0& 0& -0.0096& 0& 0\\ 0& 0& -0.01229725& 0& 0& 0& 0& 0& 0\\ 1& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 0& 0& 0\end{array}\end{array}$$

$$\begin{array}{cccccccccccccccccccccccccccccc}& & & & & & & & & & & & & & & & & & & & & \begin{array}{cccccc}0& 0& -0.00913& 0& 1& 0\\ 0& 0& 0& -0.01229& 0& 1\\ 0& 0& 0& -0.01229& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& -0.01110& -0.00965& 0& 0& 0\\ -0.01060& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0.03380& -0.01034& 0& -0.01285& 0& 0\\ -0.01034& 0.02145& 0& 0& 0& 0\\ 0& 0& 0.0294& 0& 0& 0\\ -0.01285& 0& 0& 0.0251& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\end{array}& ]& [& \begin{array}{c}{V}_1\\ V_2\\ V_3\\ V_4\\ V_5\\ V_6\\ V_7\\ V_8\\ V_9\\ V_{10}\\ V_{11}\\ V_{12}\\ V_{13}\\ i_{VE}\\ i_{VG}\end{array}& ]& =& [& \begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ V_E\\ V_G\end{array}& ]\end{array}$$

After the DC operating point is calculated by solving the system of Equations $Ax=b$, the nodal voltages and branch currents are obtained, as shown in

Table 4. Nodal voltages, resistance value, and branch currents.

Nodal Voltage (V)	Branch	Resistance	Value ($\mathbf{\Omega }$)	Current (mA)
$V\left(n_1\right)=50$	$[n_3,n_4]$	$R_1$	96.845	2.49840
$V\left(n_2\right)=0$	$[n_4,n_5]$	$R_2$	95.873	−3.03742
$V\left(n_3\right)=8.48411$	$[n_5,n_6]$	$R_3$	94.628	−57.8219
$V\left(n_4\right)=8.24215$	$[n_7,n_8]$	$R_4$	98.023	51.0919
$V\left(n_5\right)=8.53336$	$[n_8,n_4]$	$R_5$	159.817	−5.5358
$V\left(n_6\right)=14.00497$	$[n_9,n_2]$	$R_6$	102.322	224.307
$V\left(n_7\right)=12.36562$	$[n_2,n_8]$	$R_7$	101.072	−72.7935
$V\left(n_8\right)=7.35743$	$[n_8,n_{10}]$	$R_8$	94.289	−16.1658
$V\left(n_9\right)=22.95166$	$[n_{11},n_{10}]$	$R_9$	96.651	18.6642
$V\left(n_{10}\right)=8.88170$	$[n_2,n_{12}]$	$R_{10}$	94.176	−208.028
$V\left(n_{11}\right)=10.68561$	$[n_1,n_9]$	$R_{11}$	95.871	282.129
$V\left(n_{12}\right)=19.59144$	$[n_9,n_6]$	$R_{12}$	154.728	57.8219
$V\left(n_{13}\right)=8.68728$	$[n_1,n_{12}]$	$R_{13}$	109.467	277.784
$i_{VE}=-0.55991$	$[n_{12},n_7]$	$R_{14}$	103.586	69.7561
	$[n_7,n_{11}]$	$R_{15}$	90.012	18.6642
	$[n_{10},n_{13}]$	$R_{16}$	77.815	2.49840
	$[n_{13},n_3]$	$R_{17}$	81.318	2.49840
	$[n_2,n_5]$	$R_{18}$	155.762	−54.7844

. Finally, the sub-route between $n_1$ and $n_2$ is obtained using MLCCA, considering the permitted directions of vehicular flow within the road networks. Additionally, Table 4 shows the branches that form the equivalent electrical circuit of the map under analysis and their resistive values.

To determine the sub-route between $n_1$ and $n_3$, $V_G=0$ volts is placed at $n_3$. Later, matrix A is updated, then $Ax=b$ is solved again, and the nodal voltages and branch currents are recalculated. The sub-route between $n_1$ and $n_3$ is then determined using MLCCA. This process is repeated to calculate the sub-routes $n_1-n_6$ and $n_1-n_{11}$.

In

Table 5. Distances between nodes.

	${\mathit{n}}_1$	${\mathit{n}}_2$	${\mathit{n}}_3$	${\mathit{n}}_6$	${\mathit{n}}_{11}$
${\mathit{n}}_{\mathbf{1}}$	0	198.194	558.852	250.6	303.067
${\mathit{n}}_{\mathbf{2}}$	198.194	0	354.496	250.391	292.013
${\mathit{n}}_{\mathbf{3}}$	537.984	384.482	0	287.348	255.785
${\mathit{n}}_{\mathbf{6}}$	250.6	257.05	611.546	0	549.063
${\mathit{n}}_{\mathbf{11}}$	490.207	292.013	255.785	549.063	0

, all distances calculated between the target nodes are shown.

Table 6. Routes and distances obtained among $n_1$ and target nodes.

Route	Distance (m)	Route	Distance (m)
[$n_1$, $n_2$, $n_{11}$, $n_3$, $n_6$]	1033.34	[$n_1$, $n_6$, $n_3$, $n_{11}$, $n_2$]	1409.944
[$n_1$, $n_6$, $n_2$, $n_{11}$, $n_3$]	1055.448	[$n_1$, $n_6$, $n_{11}$, $n_3$, $n_2$]	1439.93
[$n_1$, $n_{11}$, $n_3$, $n_6$, $n_2$]	1103.25	[$n_1$, $n_6$, $n_{11}$, $n_2$, $n_3$]	1446.172
[$n_1$, $n_6$, $n_2$, $n_3$, $n_{11}$]	1117.931	[$n_1$, $n_{11}$, $n_2$, $n_6$, $n_3$]	1457.016
[$n_1$, $n_{11}$, $n_3$, $n_2$, $n_6$]	1193.725	[$n_1$, $n_{11}$, $n_6$, $n_2$, $n_3$]	1463.676
[$n_1$, $n_{11}$, $n_2$, $n_3$, $n_6$]	1236.924	[$n_1$, $n_6$, $n_3$, $n_2$, $n_{11}$]	1538.641
[$n_1$, $n_2$, $n_6$, $n_{11}$, $n_3$]	1253.433	[$n_1$, $n_3$, $n_{11}$, $n_6$, $n_2$]	1620.749
[$n_1$, $n_2$, $n_6$, $n_3$, $n_{11}$]	1315.916	[$n_1$, $n_2$, $n_{11}$, $n_6$, $n_3$]	1650.816
[$n_1$, $n_3$, $n_{11}$, $n_2$, $n_6$]	1357.041	[$n_1$, $n_3$, $n_6$, $n_{11}$, $n_2$]	1687.275
[$n_1$, $n_2$, $n_3$, $n_{11}$, $n_6$]	1357.538	[$n_1$, $n_3$, $n_2$, $n_6$, $n_{11}$]	1742.788
[$n_1$, $n_2$, $n_3$, $n_6$, $n_{11}$]	1389.101	[$n_1$, $n_3$, $n_2$, $n_{11}$, $n_6$ ]	1784.409
[$n_1$, $n_3$, $n_6$, $n_2$, $n_{11}$ ]	1395.263	[$n_1$, $n_{11}$, $n_6$, $n_3$, $n_2$]	1848.158

shows the possible routes and the total distance obtained, considering $n_1$ as the origin, as well as the order in which the delivery points $n_2$, $n_3$, $n_6$, and $n_{11}$ should be visited. The order in which the delivery points should be visited is: first $n_2$, then $n_{11}$, followed by $n_3$, and finally $n_6$. The distance of this route is 1033.34 m, which is the shortest among all the routes.

Figure 6 shows the route obtained with a green line. The arrows on the line indicate the direction to go from $n_1$ to the delivery points in the following order: $n_2$, $n_{11}$, $n_3$, and $n_6$. The route starts at $n_1$, with $n_2$ as the first delivery point, so the sub-route is formed by nodes $n_1$, $n_9$, and $n_2$.

The next delivery point is $n_{11}$, which is reached by a sub-route that starts at $n_2$ and passes through $n_8$ and $n_{10}$ until it reaches $n_{11}$. Subsequently, the delivery point $n_3$ is reached by a sub-route composed of $n_{11}$, $n_{10}$, and $n_3$. Finally, the last sub-route connects nodes $n_3$, $n_4$, $n_5$, and $n_6$, with $n_6$ being the final delivery point.

Therefore, the route is composed of nodes $n_1$, $n_9$, $n_2$, $n_8$, $n_{10}$, $n_{11}$, $n_{13}$, $n_3$, $n_4$, $n_5$, and $n_6$, with a return point at node $n_{11}$. It should be clarified that although in the illustrative example shown $n_1$ is chosen as the origin and $n_2$, $n_3$, $n_6$, and $n_{11}$ as delivery points, any other node can be chosen as the origin without having to recalculate distances. This saves computational resources and is a significant advantage, as it is possible to calculate a route near the shortest as well as other alternative routes.

Calculating distances between geographic points of interest within a street network is essential, as the distance and route between an origin and destination can change when the direction of travel is reversed. Factors such as street direction, traffic restrictions, one-way streets, and the location of turnaround points directly affect both the route selected and the total distance calculated. Considering all of these elements is key to optimizing route planning, ensuring that the most appropriate routes are chosen for the actual conditions of the environment.

In the following section, three case studies are presented, each applied to maps of different cities. These examples illustrate how ROMP is used for route planning in product delivery. Each case takes into account factors such as the geographical distribution of delivery points, the characteristics of road networks, and their specific restrictions.

Additionally, for each case presented with ROMP, the corresponding route was also obtained using the OpenRouteService (ORS) library, which relies on OpenStreetMap (OSM) for road network data, enabling route planning services [37]. ORS integrates GeoTools, an open-source library for geospatial data processing, and uses algorithms like Dijkstra and A* to calculate routes in road networks, optimizing path searches [37]. Real-time speed information is essential for accurate travel time estimation and route recommendations but was not considered in the case studies. ROMP and ORS rely on OSM as their primary data source, where speed limits derive from OSM tags, but real-time traffic data are not openly available. As a result, ROMP and ORS cannot determine exact travel times for a given route [38].

4. Case Studies

The ROMP method was applied to three case studies in three different cities using the $R=L$ street block length model. Additionally, for each of the cases presented with ROMP, the corresponding route was also obtained using the OpenRouteService (ORS) library, designed to calculate routes between geographic points. For better visualization, the routes are displayed in geojson.io, an online map viewer. The routes were calculated using a computer with an Intel Core i7 64-bit processor (Intel Corporation, Santa Clara, CA, USA) at 3.6 GHz and 16 GB of RAM. The software platform used is Python 3.7 on the Microsoft Windows 10 operating system.

The first case study corresponds to a map of the city of Xalapa, Veracruz, Mexico. On the map, five geographical positions are marked with icons in different colors to identify specific locations. The ROMP methodology was applied by selecting the position marked with the black icon as the origin and the green, red, yellow, and blue icons as the four delivery points. The route to the delivery points is in the following order: first, the geographic position indicated with the red icon; second, the yellow icon; then the green icon; and finally, the blue icon. Figure 7 shows the resulting route, where the arrows on the route indicate the direction to follow from the selected origin to each delivery point. The green arrows next to the red and green icons indicate the return points to the next destination. The distance obtained is $2.743$ km.

Figure 8 shows the route obtained using the OpenRouteService (ORS) library for the same map of the city of Xalapa. ORS is a library designed to determine the route between geographic points. The origin and delivery points are determined according to the order previously defined with ROMP. Although the order of delivery points remains the same, there are notable variations in the trajectory of the route calculated with ORS compared to that obtained with ROMP, as shown in Figure 8. The distance obtained was $2.810$ km. The arrows on the route indicate the path to follow from the origin.

For the second case study, a map of Cholula in the state of Puebla, Mexico, was chosen. The map features six geographical positions marked with icons in different colors. The origin of the route is the green icon, while the delivery points are represented by the black, red, yellow, navy blue, and gray icons. Figure 9 shows the route, which spans a distance of 3.281 km. The arrows on the route indicate the direction to follow to each of the destinations.

Figure 10 shows the route obtained using ORS for the same map of the city of Cholula. Each sub-route was calculated using the ORS library, and the overall route follows the same order as with ROMP. In Figure 10, we observe that four of the sub-routes match those obtained with ROMP; however, there is a variation of one block in the sub-route between destinations 2 and 3. The distance obtained with ORS was 3.399 km. The difference between ROMP and ORS in the total distance is 118 m, as ROMP prioritizes shorter blocks.

The third case study features a map of London, England. There are seven geographical positions marked with colored icons: gray, navy blue, blue, red, black, yellow, and green. The gray icon marks the origin of the route, while the remaining six icons indicate the destinations to be visited. Figure 11 depicts the resulting route obtained using ROMP, which covers a distance of 5.908 km. The green arrows next to the blue, red, and black icons indicate return points to the next destination.

Figure 12 shows the route obtained using ORS for the same map of the city of London, England. The sub-routes follow the same order as in ROMP, with each sub-route obtained individually using the ORS library. As shown in Figure 12, five sub-routes are identical to those obtained with ROMP, with variations only in the sub-route between Destinations 5 and 6. The distance obtained with ORS is $6.030$ km. Consequently, the route distance obtained with ROMP is shorter than that of ORS by 92 m.

In this section, three cases of vehicle routing considering one origin and multiple delivery points or destinations have been compared using two methodologies: ROMP and ORS. The results obtained show that in all cases, the ROMP methodology produced routes with a shorter travel distance. This is because it prioritizes the shortest distance blocks when calculating routes between a desired origin and destination, which is useful and has potential applications in the areas of logistics and transportation.

5. Discussion and Numerical Comparisons

This paper describes the Routing Optimization for Multiple Delivery Points (ROMP) methodology in the context of a vehicle route planning scheme that considers one origin and multiple destinations. The road network is represented as a linear electric circuit. The three case studies described in Section 4 demonstrate the capabilities of ROMP under the proposed scheme using the road length model $R=L$. According to the distances obtained in each case using ROMP, priority is given to the shortest distance, even if this means using residential or tertiary roads, which are usually avoided in commercial navigation systems. This approach allows the exploration of a new route planning scheme that can be used for the delivery of goods, medicines, and the transfer of patients to different hospitals, among other applications.

The case studies in Section 4 show that ROMP can be used to propose route planning schemes focused on the delivery of products for specific needs.

Table 7. Route search time and distance traveled.

Case	City	Method	Distance (km)	Time (ms)
1	Xalapa, Veracruz	ROMP	2.743	1.246
1	Xalapa, Veracruz	ORS	2.810	113.1
2	Cholula, Puebla	ROMP	3.281	4.255
2	Cholula, Puebla	ORS	3.399	130.9
3	London, England	ROMP	5.908	7.930
3	London, England	ORS	6.030	137.8

shows a comparison between ROMP and OpenRouteService (ORS) for each of the case studies. The parameters considered for this comparison are the route search time and the distance obtained. OpenRouteService is an open-source, open-access routing service based on OpenStreetMap (OSM) data. The services provided include route planning using OSM road network data and planning functions using various routing libraries such as GeoTools, pgRouting, and algorithms such as Dijkstra and A*.

To obtain each of the routes and distances with OpenRouteService, a code was programmed in Python 2.7 using the Folium and OpenRouteService libraries. Within the code, the starting point and destination were indicated using latitude and longitude coordinates. Subsequently, the route request was made using ORS.Client, and a list was obtained with the geographical points that make up each of the corresponding sub-routes, as well as their distance using the Haversine formula. To obtain the time used by each method, from the permutations obtained, the route with the shortest distance and the order in which the route should be made was chosen. Therefore, the time obtained for ORS is the result of the sum of the times to go from an origin to a destination; that is to say, the times used in each sub-route were added in the same way they were obtained for ROPM. Both methods, ORS and ROMP, were executed on the same computer under the same conditions. The method was executed locally, i.e., the map data download and the route calculation were performed in the same environment. Therefore, both ROMP and ORS only measure the time it takes to search each sub-route between origin and destination, thus guaranteeing identical conditions for both methodologies, eliminating the variability of network latency.

In the three cases presented with ROMP and ORS in Table 7, differences are observed in the routes obtained. In case 1, the sub-routes that make up the route are different for each of the methods shown. In case 2, the only difference is found in a street block of the sub-route formed by Destination 2 ⇒ Destination 3. Finally, in case 3, the only difference is in the sub-route formed by Destination 5 ⇒ Destination 6. As shown in the table, for each of the cases, the routes obtained using ROMP are shorter than those obtained using ORS. However, the main advantage of ROMP is its shorter route search time compared to OpenRouteService. This reduction in search time is due to the fact that ROMP only performs comparisons. Therefore, in this proposed scheme, it can be concluded that ROMP is faster in route search than ORS.

Future work should explore improved models that account for street classification, primary, secondary, tertiary, lane count, road surface type, and other data from OSM. This would enable route optimization based on preferred road types, varying traffic scenarios, and infrastructure conditions. Additionally, combining ROMP with heuristic or optimization algorithms could enhance validation. Further research is also needed to compare ROMP with other routing solvers other than ORS. Moreover, real-time management of traffic intensity and its impact on fuel consumption and travel times should be considered. These advancements could contribute to more sustainable urban mobility by improving efficiency in medicine, food, and patient transport while reducing emissions and resource consumption.

6. Conclusions

By modeling urban street networks as analog electric circuits, our approach offers a novel scheme for optimizing vehicle routing. Transforming a road network into an electrical circuit and applying Modified Nodal Analysis (MNA) creates a unique analogy between vehicle flow and electric current, opening new opportunities for network modeling and route optimization for different fields such as electrical engineering, scheduling, and logistics. Our method, ROMP, exceeds conventional solutions like OpenRouteService (ORS) by reducing both the calculation time and the total distance traveled. Case studies in Xalapa, Cholula, and London demonstrate that ROMP can compute routes in milliseconds—much faster than ORS, which takes seconds—while providing shorter routes when considering one origin and multiple destinations.

This study has important theoretical implications. By associating street block length with resistance R (see Equation (1)) and using MNA, we establish a clear relationship between physical road attributes and vehicular flow. The experimental results (see Figure 3 and Figure 6 and Table 7) provide a robust model that connects circuit theory with route optimization, facilitating future research that may integrate additional real-world parameters such as traffic congestion and road conditions.

From a managerial perspective, ROMP offers benefits for logistics and urban mobility. The reduction in route search time translates into lower operating costs and improved fuel efficiency, which is critical in high-demand situations like emergency medicine, food distribution, and supply chain management. Moreover, the method’s ability to dynamically change the origin or delivery points without recalculating all inter-node distances gives managers the flexibility needed to adapt quickly to changing conditions.

The environmental benefits are straightforward: by reducing travel distances, ROMP decreases fuel consumption and greenhouse gas emissions, contributing to more sustainable logistics. This reduction in travel not only lowers the carbon footprint but also improves the quality of life in densely populated cities by promoting sustainable mobility.

Although the street length model presented here does not currently include variables such as real-time traffic, traffic light synchronization, road infrastructure details, number of lanes, or pavement type, these parameters could be incorporated in future work to develop a more practical schema. Future directions for this research include integrating real-time data, street classifications, and vehicle-specific information, as well as exploring models that combine ROMP with machine learning techniques for predictive route optimization. In this way, ROMP has the potential to further reduce fuel consumption and environmental impact while ensuring timely delivery of products, adhering to sustainability objectives and contributing to the field of logistics.