Optimal Weigh-in-Motion Planning for Multiple Stakeholders

Abstract

Overloaded trucks contribute heavily to road damage and increased maintenance costs, and Weigh-In-Motion (WIM) systems are an effective tool for detecting them without disrupting traffic flow. However, overloaded truck drivers often adjust their routes to avoid WIM stations, complicating enforcement efforts for road management stakeholders. To address these challenges, this study integrates the strategic behaviors of multiple stakeholders with diverse objectives into optimal WIM planning by modeling interactions among the government, pavement management agencies, and drivers. The authorities are responsible for WIM installation, while drivers minimize their respective travel costs. The proposed approach considers both road maintenance costs incurred by authorities and travel costs for drivers, based on a traffic assignment model for each WIM installation strategy. Basic concepts from game theory are adopted to formalize the dynamic interactions among these stakeholders.

1. Introduction

Overloaded trucks are major contributors to increased road maintenance costs and infrastructure damage. Their excessive weight causes structural distress in road pavements, reducing their service life. Overloaded trucks can increase the overall costs of pavement maintenance compared to the same vehicles carrying legal loads [1]. In South Korea, research has shown that overloaded trucks are responsible for an estimated 36% reduction in the total lifespan of roads [2]. This issue is not limited to these regions; it is a widespread problem faced by countries around the world [3,4,5].

Given these challenges, implementing enforcement strategies to effectively reduce the number of overloaded trucks is essential. Common enforcement approaches include stationary and mobile weigh stations. Stationary weigh stations utilize high-speed and low-speed axle load scales to monitor overloaded trucks at fixed locations. In contrast, mobile enforcement employs portable axle load scales at temporary inspection points.

Weigh-In-Motion (WIM) is a type of stationary weigh station system that measures vehicle weight and axle loads using sensors embedded in the road. WIM systems have the advantage of enforcing regulations without disrupting traffic flow, as vehicles do not need to stop or slow down. Additionally, WIM collects valuable data automatically, including truck type, axle loads, and speed, which can be integrated with driver information [6]. Its effectiveness in mitigating road damage caused by overloaded trucks has been well documented due to its efficiency. For example, implementing WIM enforcement on heavily damaged road sections in Montana led to a reduction in road damage costs by approximately USD 700,000 annually [7].

Although WIM offers efficient data collection without significantly affecting traffic flow, it is impractical to install WIM systems on every road. Moreover, overloaded truck drivers, using their experience, may deliberately avoid WIM stations. In one study, 11–14% of all trucks, consisting primarily of overloaded vehicles, intentionally bypassed inspection stations [8]. Therefore, it is crucial to strategically determine WIM locations to monitor as many overloaded trucks as possible and promote compliance with legal loading practices.

Selecting the optimal locations for WIM installations can be formulated as an example of the Evasive Flow Capturing Problem (EFCP), which considers drivers’ tendency to change routes to avoid inspection facilities. Previous research explored how to minimize damage costs from overloaded trucks by optimizing the location of inspection stations using pre-generated routes and binary integer programming [6]. Other studies addressed multi-period stochastic EFCP [9], and bilevel programs without predetermined routes, applying duality theory to convert the problem into a single-stage program, which was solved using branch-cut algorithms [10]. Further research incorporated a pessimistic approach, assuming drivers have limited rationality and use general cost functions to make decisions [11].

Many of these studies assume that drivers are willing to take detours if the new route’s costs or travel time is within a set tolerance compared to their original path [6,9,10,11]. For instance, when a driver’s shortest route is blocked by a WIM station, they might take a detour if the time or distance penalty is acceptable. However, in real-world situations, drivers also consider whether avoiding fines for overloading is worth the extra cost of taking longer detours that go beyond their tolerance. As a result, some drivers might opt to carry non-overloaded loads, shifting overloaded demand into non-overloaded demand. Existing studies do not fully capture this behavioral complexity.

To address these limitations and more accurately capture the strategic behavior of overloaded truck drivers, game theory serves as an effective tool for analyzing interactions between decision-makers in this context. Game theory primarily studies conflicts and interactions among multiple decision-makers to determine optimal strategies, taking into account both internal and external processes, whether at the same or different levels. In the transportation field, the main players in game theory are authorities and travelers. Game theory has been widely applied to resolve various network-level issues in games between authorities and travelers, including toll policies [12,13,14], vehicle routing problems [15], and transportation network reliability [16]. Some studies have also applied game theory to pavement maintenance management [17], focusing on interactions between agencies and service providers responsible for actual maintenance activities. These studies propose pavement maintenance strategies for multiple decision-makers.

However, road maintenance activities and the installation of WIM systems not only affect the flow of overloaded trucks but also influence other road users. Therefore, it is crucial to consider both authorities and road users in this interaction. Authorities must operate within a fixed budget to carry out road maintenance and install WIM systems, while drivers respond to policies by interacting with one another to minimize their own travel costs. This interaction can be modeled as a Stackelberg game, a framework for hierarchical decision-making. In such a game, the leader makes the first move, and the followers respond strategically. This sequential process enables the leader to anticipate the followers’ reactions and optimize their strategy accordingly. In this study, the government or agency acts as the leader, making the first move by determining the locations for WIM installations within a fixed budget. Multi-type drivers, including overloaded truck drivers, act as followers, adjusting their routes and behavior in response to the enforcement strategies. By modeling this interaction as a Stackelberg game, this study captures the hierarchical nature of decision-making and the strategic interplay between authorities and road users, allowing for an optimized policy design.

Furthermore, among drivers, a Nash game is formed, where each driver independently chooses their optimal route while considering the choices made by other drivers. In a Nash game, which models strategic interactions among rational decision-makers, each participant selects their strategy to maximize their own benefit or minimize their costs, given the strategies of others. This leads to a Nash equilibrium, a stable state where no participant can improve their outcome by unilaterally changing their strategy. In the context of this study, drivers aim to minimize their travel costs while accounting for others’ choices. Moreover, in this game, we assume that if the costs of taking an evasive detour for an overloaded truck driver is higher than the costs of giving up overloading, they will stop overloading, which can affect the choices of other drivers as well. These interactions between drivers can influence how overloaded trucks distribute themselves across the network in response to WIM installations and enforcement strategies.

By incorporating game theory into this context, this study aims to provide a more realistic and comprehensive model of how authorities can optimize WIM placement and enforcement strategies while considering the strategic behavior of overloaded truck drivers and the interactions among road users.

2. Stakeholders and Game Scenarios

Placing WIM stations can result in converting overloaded trucks to non-overloaded trucks, redistributing traffic loadings over road networks, and potentially increasing traffic disruptions. In WIM placement planning, there are three groups of stakeholders: (i) the government, which aims to minimize both travel time costs and pavement management costs; (ii) the roadway management agency, which primarily focuses on pavement management costs under the minimum required pavement serviceability constraint; and (iii) roadway users, including light-duty vehicles (i.e., regular vehicles), non-overloaded trucks, and overloaded trucks, who aim to minimize their individual travel costs or maximize their travel benefits. The objectives of different stakeholders vary, and while a WIM placement may benefit one stakeholder, it might not be advantageous for others. To consider such complex dynamics among multiple stakeholders in WIM planning, it is necessary to systematically identify the ecosystem of the stakeholders in terms of the different impacts of WIM planning on each of them. We assume that the government requires the pavement management agency to keep the condition of each pavement section within a certain range (e.g., with the maximum allowable roughness in the International Roughness Index (IRI) [m/km]).

First scenario: we assume that the government controls the pavement management budget for the agency, and its objective is to minimize the summation of the user costs and agency’s pavement management costs by determining the optimal WIM placement. Each driver in the three user groups tries to minimize their travel costs. Given WIM placement, we can interpret the situation as the drivers playing a Nash game while the government and the user groups play a bi-level Stackelberg game as the leader and followers as shown in Figure 1a. Note that the agency is not an independent game player since it is directly controlled by the government.

Second scenario: based on the IRI trigger given by the government for the minimum required service level for users, the agency determines their optimal WIM placement to minimize the pavement management costs for pavement managerial activities through its redistribution and reduction in concentrated traffic loading. Additionally, the government can impose limits on network-wide traffic disruptions resulting from the agency’s WIM implementation within a specific level. This game is also formulated as a bi-level Stackelberg game, in which the agency acts as the leader and the users as followers, as shown in Figure 1b. At the lower level, users engage in a Nash game.

3. Formulations

All the notations used in this section are presented in Appendix A. We analyze roadway network graph $G=(N,A)$, comprising nodes N and arcs A. The nodes and arcs are indexed by $n$ and $a$, respectively, and each arc $a$ has a length denoted by $l_a$ [km]. A WIM system can be installed on certain candidate arcs, denoted by $A^{\prime }$. The decision variable is the set of arcs where WIM is installed, represented as $W\left(\subseteq A^{\prime }\right)$. Accordingly, we define binary variables $w_a$, where $w_a=1$ indicates that a WIM is implemented on candidate arc $a$, and $w_a=0$ otherwise.

We have origin–destination (O-D) matrices available before the implementation of any WIM system (i.e., when $W=\mathbf{0}$, representing an empty set). These matrices are denoted as $q^i\left(\mathbf{0}\right)$ $\left(\in {\mathbb{R}}_{\ge 0}^{\left|N\right|\times \left|N\right|}\right)$ for different vehicle types: overloaded trucks ($i=1$), non-overloaded trucks ($i=2$), and regular vehicles ($i=3$), respectively. For the O-D pair from node $n$ to node $n^{\prime }$, the demand for each of three vehicle types is represented by $q_{{nn}^{\prime }}^i\left(\mathbf{0}\right)$. After the WIM strategy is implemented (i.e., $W\ne \mathbf{0}$), the demands for vehicle types are adjusted and denoted as $q^i\left(W\right)$ for all $i$.

The total flow rate on link $a$ is $x_a\equiv \left\{x_a^i,\forall i\right\}$, where the flow rate of vehicle type $i$ on link $a$ is $x_a^i$. The travel time on link $a$ for vehicle type i is $t_a^i$, which is an increasing function of $x_a^i$ for all $i$. The network flow, defined as $x\equiv \left\{x_a,\forall a\right\}$, and each link flow $x_a$ are influenced by WIM strategy $W$, so they are expressed as $x\left(W\right)$ and $x_a\left(W\right)$, respectively.

3.1. Upper-Level Problem (Leader’s Strategy)

3.1.1. First Scenario

The upper-level leader, the government, directly decides the WIM placement $W$ with the objective to minimize the summation of travel costs $T_a$ [USD/h] and pavement management costs $A_a$ [USD/h] for all links $a$, where $T_a$ and $A_a$ are increasing functions of traffic volumes on link $a$. The non-negative and non-dimensional weight factor $\omega$ is introduced to incorporate the problem into a bi-criteria framework. Note that the link-specific traffic volumes are derived from $x_a$, which varies depending on the decision $W$.

$$\underset{W}{\min }\sum _a\left[T_a\left(x_a\left(W\right)\right)+\omega A_a\left(x_a\left(W\right)\right)\right]$$

(1)

The travel cost $T_a$ is

$$T_a=\sum _i\left[{\sigma }^i{x}_a^i{t}_a^i\left(x_a\right)+l_a{\gamma }^i{x}_a^i\right]$$

(2)

where ${\sigma }^i$ [USD/h/veh] represents the average unit travel time costs per vehicle for vehicle type $i$, excluding fuel-related costs; $t_a^i\left(x_a\right)$ is the travel time of link $a$ for vehicle type $i$; and ${\gamma }^i$ is fuel efficiency costs for vehicle type $i$ [USD/km]. As mentioned earlier, $t_a^i\left(x_a\right)$ is an increasing function of traffic volumes on the link, such as the BRP function presented in Equation (4):

$$x_a^i=\sum _n\sum _{n^{\prime }}\sum _k{f}_{n{n}^{\prime },k}^i{\delta }_{n{n}^{\prime },a,k}^i$$

(3)

$$t_a^i\left(x_a\right)={\mathcal{t}}_a^i\left\{1+0.15{\left({\displaystyle \frac{\sum _{i^{\prime }}{\alpha }^{i^{\prime }}{x}_a^{i^{\prime }}}{{\kappa }_a}}\right)}^{\phi }\right\}$$

(4)

where $k$ is the path between OD pair $n$ to $n^{\prime }$, and ${\delta }_{n{n}^{\prime },a,k}^i$ is the indicator variable defined as 1 if link $a$ is included in path $k$ between $n$ and $n^{\prime }$ for vehicle type $i$ and is 0 otherwise; $f_{n{n}^{\prime },k}^i$ [veh/h] is the flow on path $k$ for vehicle type $i$ between $n$ and $n^{\prime }$; ${\mathcal{t}}_a^i$ [h] is the free-flow travel time on link $a$ for vehicle type $i$; ${\kappa }_a$ [veh/h] stands for the link capacity; and ${\alpha }^{i^{\prime }}$ and $\phi$ are positive coefficients. We assume that the free-flow speed and the impact on the traffic environment of non-overloaded and overloaded trucks are the same and higher than those of light-duty vehicles, i.e., ${\mathcal{t}}_a^1={\mathcal{t}}_a^2>{\mathcal{t}}_a^3$, and ${\alpha }^1={\alpha }^2>{\alpha }^3$.

If we consider undiscounted prorated pavement management costs with single rehabilitation costs on link $a$, denoted by $M_a$, the pavement management costs $A_a$ is as follows:

$$A_a={\displaystyle \frac{M_a}{{\tau }_a}}$$

(5)

where ${\tau }_a$ is the steady-state rehabilitation period of link $a$. Any functional form of a deterioration model for pavement condition can be applied; therefore, we use the following function with the traffic loading term, ${TL}_a$, quantified in Equivalent Single-Axle Loadings (ESALs), which is a slightly simplified version of that from [18]:

$$s_a\left(\mu \right)=s^{+}{e}^{\left(b_0+b_1{TL}_a\right)\mu }$$

(6)

where $\mu$ means the elapsed time since the last rehabilitation; $s_a\left(\mu \right)$ is the pavement condition of link $a$ at $\mu$, such as in [m/km] if the condition is defined with the International Roughness Index (IRI); $s^{+}$ is the condition right after rehabilitation; $b_0$ is a positive parameter in [h⁻¹]; and $b_1$ is a positive parameter in [(ESALs×h)⁻¹]. The traffic loading is a linear summation of traffic volumes with vehicle-group-specific parameters ${\lambda }_i$:

$${TL}_a=\sum _i{\lambda }_i{x}_a^i$$

(7)

Given a predetermined pavement condition threshold $s^{-}$ for initiating a rehabilitation activity, the rehabilitation period is expressed as follows:

$${\tau }_a={\displaystyle \frac{\ln s^{-}-\ln s^{+}}{b_0+b_1{TL}_a}}$$

(8)

Finally, the pavement management costs $A_a$ is derived as a linear function of ${TL}_a$:

$$A_a={\displaystyle \frac{M_a\left(b_0+b_1{TL}_a\right)}{\ln s^{-}-\ln s^{+}}}$$

(9)

In summary, the upper-level game is given as the following Mathematical Programming with the WIM number constraint limited by $B$, which indicates a separate budget for WIM installation:

$$\underset{W}{\min }\sum _a\left[\sum _i{\sigma }^i{x}_a^i{\mathcal{t}}_a^i\left[1+0.15{\left({\displaystyle \frac{\sum _{i^{\prime }}{\alpha }^{i^{\prime }}{x}_a^{i^{\prime }}}{{\kappa }_a}}\right)}^{\phi }\right]+l_a{\gamma }^i{x}_a^i+\omega {\displaystyle \frac{M_a\left(b_0+b_1\sum _{i^{\prime }}{\lambda }_{i^{\prime }}{x}_a^{i^{\prime }}\right)}{\ln s^{-}-\ln s^{+}}}\right]$$

(10)

subject to

$$\left|W\right|\le B$$

(11)

3.1.2. Second Scenario

The upper-level leader, the pavement management agency, determines $W$ to minimize the pavement management costs $A_a$ [USD/h] for all links $a$ as shown in Equation (12), with the two user-related constraints given by the government. For the first constraint, the roughness threshold is already defined as $s^{-}$ in the calculation of $A_a$. The second constraint with the maximum allowable disruption level $D$ [veh-h/h] is addressed as Equation (13), where the second term in the left-hand side refers to the original vehicle-hours traveled per hour without WIM implemented, where $W=\mathbf{0}$. Finally, the separate budget constraint for the WIM number is considered.

$$\underset{W}{\min }\sum _a{A}_a\left(x_a\left(W\right)\right)$$

(12)

s.t.

$$\sum _a{T}_a\left(x_a\left(W\right)\right)-\sum _a{T}_a\left(x_a\left(\mathbf{0}\right)\right)\le D$$

(13)

$$\left|W\right|\le B$$

(14)

3.2. Lower-Level Problem (Follower’s Strategy)

The lower-level problem involves finding path and link flows for all vehicle types based on $W$ from the upper-level problem. As is generally assumed, truck drivers and regular light-duty vehicle drivers select their paths to minimize travel costs, including time and fuel cost.

For the original overloaded truck drivers with a demand of $q^1\left(\mathbf{0}\right)$, the overloading benefit is defined as the cost difference for an overloaded truck driver making a one-way trip between its origin and destination along the lowest-cost path, compared to the cost of a non-overloaded truck on its lowest-cost path, in the absence of WIM installation. Here, the lowest-cost path travel time and distance for overloaded trucks between origin node $r$ to destination node $p$ under $W=\mathbf{0}$ are $t_{rp}^1{\left(\mathbf{0}\right)}^{*}$ and $d_{rp}^1{\left(\mathbf{0}\right)}^{*}$. Those for non-overloaded trucks and regular vehicles are ($t_{rp}^2{\left(\mathbf{0}\right)}^{*}$, $d_{rp}^2{\left(\mathbf{0}\right)}^{*}$) and ($t_{rp}^3{\left(\mathbf{0}\right)}^{*}$, $d_{rp}^3{\left(\mathbf{0}\right)}^{*}$). In summary, overloading occurs between OD of $r$ and $p$ because of the following:

$${\sigma }^1{t}_{rp}^1{\left(\mathbf{0}\right)}^{*}+{\gamma }^1{d}_{rp}^1{\left(\mathbf{0}\right)}^{*}-{\epsilon }_{rp}<{\sigma }^2{t}_{rp}^2{\left(\mathbf{0}\right)}^{*}+{\gamma }^2{d}_{rp}^2{\left(\mathbf{0}\right)}^{*}$$

(15)

where ${\epsilon }_{rp}$ is the extra income from overloading between $r$ and $p$. The overloading benefit, denoted as $h_{rp}\left(\mathbf{0}\right)$, equals the left-hand side (LHS) minus the right-hand side (RHS) of the above inequality. However, even if the overloading benefit is positive between the OD of $r$ and $p$, not all trucks for this OD will overload since it is illegal. Thus, the maximum bound of overloaded truck rate is set at $q^1\left(\mathbf{0}\right)$.

The overloading penalty $\Delta$ occurs when an overloaded truck passes a WIM. It is assumed that the overloading penalty is significantly higher than the overloading benefit, i.e., $\Delta \gg h_{rp}$ for all pairs of $r$ and $p$. The cost to travel arc $a$ with WIM for overloaded trucks is ${\sigma }^1{t}_a^1\left(W\right)+{\gamma }^1{l}_a+\Delta$, while that for non-overloaded trucks is ${\sigma }^2{t}_a^2\left(W\right)+{\gamma }^2{l}_a$ and for regular vehicles is ${\sigma }^3{t}_a^3\left(W\right)+{\gamma }^3{l}_a$. These travel times can be directly determined by $x_a=\left\{x_a^1,x_a^2,x_a^3\right\}$, and they are eventually indirect functions of $W$.

If the cost of overloading drivers on the new optimal path after implementing $W$ with path time and distance $t_{rp}^1{\left(W\right)}^{*}$ and $d_{rp}^1{\left(W\right)}^{*}$ is higher than the new optimal path when not overloading, the overloaded truck drivers will refuse to overload. Between $r$ and $p$, the number of reduced overloaded trucks $y_{rp}$ increases from zero until overloading brings a benefit or until $y_{rp}$ becomes $q_{rp}^1\left(\mathbf{0}\right)$ (i.e., all overloaded trucks have transitioned to non-overloaded trucks). We consider the conversion factor $\eta$ between the increased number of non-overloaded trucks when a single overloaded truck is reduced, which is higher than 1. If the overloading reduction is given by $y=\left\{y_{rp},\forall r,p\right\}$, the non-refusal (keeping overloading) condition is given as follows:

$${\sigma }^1{t}_{rp}^1{\left(W,y\right)}^{*}+{\gamma }^1{d}_{rp}^1{\left(W,y\right)}^{*}-{\epsilon }_{rp}<{\sigma }^2{t}_{rp}^2{\left(W,y\right)}^{*}+{\gamma }^2{d}_{rp}^2{\left(W,y\right)}^{*}$$

(16)

Similarly, the benefit of overloading $h_{rp}\left(W,y\right)$ is the difference between the LHS and the RHS of the above inequality.

The original graph $G$ is augmented into a super-network $G^{\prime }$ with three layers of $G^1=\left(N^1,A^1\right)$, $G^2=\left(N^2,A^2\right)$, and $G^3=\left(N^3,A^3\right)\left(=G\right)$, corresponding to overloaded trucks, non-overloaded trucks, and regular vehicles, respectively. For each truck OD pair of $r$ and $p$, the corresponding origin nodes $r^1\in N^1$ for an overloaded truck and $r^2\in N^2$ for a non-overloaded truck are connected by dummy node $r^{truck}$. The destination nodes are also connected to $p^{truck}$. We define $\left(N^1,A^1\right)=\left(N+\left\{r^{truck},p^{truck},\forall r,p\right\},A+\left\{\overrightarrow{r^{truck}{r}^1},\overrightarrow{p^1{p}^{truck}},\forall r,p\right\}\right)$ and $\left(N^2,A^2\right)=\left(N+\left\{r^{truck},p^{truck},\forall r,p\right\},A+\left\{\overrightarrow{r^{truck}{r}^2},\overrightarrow{p^2{p}^{truck}},\forall r,p\right\}\right)$, where $\overrightarrow{r^{truck}{r}^1}$ is a directed link between nodes $r^{truck}$ and $r^1$. The links connected with the dummy nodes are associated with zero travel costs, except for $\overrightarrow{r^{truck}{r}^1}$ having $-{\epsilon }_{rp}$. On $G^3$, the corresponding nodes to $r$ and $p$ are denoted as $r^3$ and $p^3$, respectively, without a dummy. For example, a simple network comprising two nodes and one directed link $a$ with a single OD pair for trucks has the super-network graphically described as described in Figure 2:

The links of $G^1$, $G^2$, and $G^3$ corresponding to $a\in A$ are denoted by $a^1$, $a^2$, and $a^3$, respectively. These links are mutually independent with asymmetric interaction. The link flows in the super-network $\left(x_{a^1},x_{a^2},x_{a^3}\right)$ are equal to $\left(x_a^1,x_a^2,x_a^3\right)$, and the link set of WIM installed on $G^1$, denoted by $W^1$, is identical to $W$. The monetized link travel costs are as follows:

<for overloaded truck network $G^1$>

$$m_{a^1}\left(x_{a^1},x_{a^2},x_{a^3}|W^1\right)={\sigma }^1{t}_a^1\left(x_{a^1},x_{a^2},x_{a^3}\right)+{\gamma }^1{l}_a,\forall a^1\in A^1-W^1$$

(17)

$$m_{a^1}\left(x_{a^1},x_{a^2},x_{a^3}|W^1\right)={\sigma }^1{t}_a^1\left(x_{a^1},x_{a^2},x_{a^3}\right)+{\gamma }^1{l}_a+\Delta ,\forall a^1\in W^1$$

(18)

$$m_{\overrightarrow{r^{truck}{r}^1}}=-{\epsilon }_{rp},\forall r,p$$

(19)

$$m_{\overrightarrow{p^1{p}^{truck}}}=0,\forall p$$

(20)

<for non-overloaded truck network $G^2$>

$$m_{a^2}\left(x_{a^1},x_{a^2},x_{a^3}\right)={\sigma }^2{t}_a^2\left(x_{a^1},x_{a^2},x_{a^3}\right)+{\gamma }^2{l}_a,\forall a^2\in A^2$$

(21)

$$m_{\overrightarrow{r^{truck}{r}^2}}=m_{\overrightarrow{p^2{p}^{truck}}}=0,\forall r,p$$

(22)

<for regular vehicle network $G^3$>

$$m_{a^3}\left(x_{a^1},x_{a^2},x_{a^3}\right)={\sigma }^3{t}_a^3\left(x_{a^1},x_{a^2},x_{a^3}\right)+{\gamma }^3{l}_a,\forall a^3\in A^3$$

(23)

According to the mathematical model for super-networks with interdependent mixed traffic [19], the user equilibrium is found by solving the following assignment problem for given $W$:

$$\begin{gathered} \underset{f|W}{\min }z\left(f|W\right)=\sum _{a^1\in A^1}{\int }_0^{x_{a^1}}{m}_{a^1}\left(\xi ,x_{a^2},x_{a^3}|W^1=W\right)d\xi +\sum _{a^2\in A^2}{\int }_0^{x_{a^2}}{m}_{a^2}\left(x_{a^1},\xi ,x_{a^3}\right)d\xi \\ +\sum _{a^3\in A^3}{\int }_0^{x_{a^3}}{m}_{a^3}\left(x_{a^1},x_{a^2},\xi \right)d\xi \end{gathered}$$

(24)

s.t.

$$\sum _k{f}_{r^{truck}{p}^{truck},k}^1+\eta \sum _k{f}_{r^{truck}{p}^{truck},k}^2=q_{rp}^1\left(\mathbf{0}\right)+\eta q_{rp}^2\left(\mathbf{0}\right),\forall r,p$$

(25)

$$\sum _k{f}_{r^{truck}{p}^{truck},k}^1\le q_{rp}^1\left(\mathbf{0}\right),\forall r,s$$

(26)

$$\sum _k{f}_{r^{car}{p}^{car},k}^3=q_{rp}^1\left(\mathbf{0}\right),\forall r,p$$

(27)

$$f_{r^{truck}{p}^{truck},k}^1,f_{r^{truck}{p}^{truck},k}^2,f_{rp,k}^3\ge 0,\forall k,r,p$$

(28)

$$x_{a^1}=\sum _{r^{truck}}\sum _{p^{truck}}\sum _k{f}_{r^{truck}{p}^{truck},k}^1{\delta }_{a^1,r^{truck}{p}^{truck},k}^1$$

(29)

$$x_{a^2}=\sum _{r^{truck}}\sum _{p^{truck}}\sum _k{f}_{r^{truck}{p}^{truck},k}^2{\delta }_{a^2,r^{truck}{p}^{truck},k}^2,$$

(30)

$$x_{a^3}=\sum _{r^{car}}\sum _{p^{car}}\sum _k{f}_{rp,k}^3{\delta }_{a^3,r^{car}{p}^{car},k}^3$$

(31)

where $f_{r^{truck}{p}^{truck},k}^1$, $f_{r^{truck}{p}^{truck},k}^2$, and $f_{rp,k}^3$ are the flows on path $k$ on each corresponding graph between OD $r$ and $p$ for three vehicle groups ($f=\left\{f^1,f^2,f^3\right\}$; $f^1=\left\{f_{r^{truck}{p}^{truck},k}^1,\forall r,p,k\right\}$, $f^2=\left\{f_{r^{truck}{p}^{truck},k}^2,\forall r,p,k\right\}$, and $f^3=\left\{f_{r^{car}{p}^{car},k}^3,\forall r,p,k\right\}$). The link flow $x=\left\{x^1,x^2,x^3\right\}$ ($x^1=\left\{x_{a^1},\forall a^1\in A^1\right\}$, $x^2=\left\{x_{a^2},\forall a^2\in A^2\right\}$, and $x^3=\left\{x_{a^3},\forall a^3\in A^3\right\}$) is a function of $f$ as $x\left(f\right)$. Equation (23) is to satisfy the Wardrop equilibrium conditions, and Constraints (24)–(30) ensure flow conservation and non-negativity. Particularly, Constraint (25) limits the maximum number of overloaded trucks for each OD. Conventional convex solution methodologies for the user equilibrium (UE) cannot be utilized for this problem due to its asymmetric interactions.

For a given $W$, we instead present an iterative algorithm to find the best $x\left(W\right)$ in Algorithm 1.

Algorithm 1. Numerical algorithm to find the user equilibrium for given WIM strategy $W$.

Input: $W$, $q^1\left(\mathbf{0}\right)$, $q^2\left(\mathbf{0}\right)$, $q^3\left(\mathbf{0}\right)$, and the other parameters Initialization: Find a feasible ${\left[x\right]}^n$ for $x$. Set $n\leftarrow 0.$

Step 1. Solve the above problem with the pre-set ${\left[x\right]}^n$ using the following objective:        $$\begin{gathered} \underset{f|W}{\mathit{min}}z\left(f\right|W)={\displaystyle \sum _{a^1\in A^1}}{\int }_0^{x_{a^1}}{m}_{a^1}\left(\xi ,{\left[x_{a^2}\right]}^n,{\left[x_{a^3}\right]}^n|W^1=W\right)d\xi +{\displaystyle \sum _{a^2\in A^2}}{\int }_0^{x_{a^2}}{m}_{a^2}\left({\left[x_{a^1}\right]}^n,\xi ,{\left[x_{a^3}\right]}^n\right)d\xi \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\displaystyle \sum _{a^{car}\in A^{car}}}{\int }_0^{x_{a^3}}{m}_{a^3}\left({\left[x_{a^1}\right]}^n,{\left[x_{a^2}\right]}^n,\xi \right)d\xi \end{gathered}$$ Step 2. The resulting $x$ with the optimal $f$ from step 1 yields ${\left[x\right]}^{n+1}.$ If $f$ converges terminate. Otherwise, set $n\leftarrow n+1$ and go to Step 1.

Output: $x$ and $f$

4. Case Study

This study utilizes the Nguyen–Dupuis Network as a case study. As illustrated in Figure 3, the network comprises 13 nodes and 19 links. The origin nodes are 1 and 4, and the destination nodes are 2 and 3, resulting in a total of four OD demand pairs. The network’s capacity (${\kappa }_a$), length ($l_a$), and other parameters are predefined, with detailed values provided in

Table 1. Nguyen–Dupuis network information.

Origin	Destination	Link	$\mathbf{Capacity}{\mathit{\kappa }}_{\mathit{a}}$ (veh/h)	$\mathbf{Length}{\mathit{l}}_{\mathit{a}}$ (km)
1	5	1	300	7
1	12	2	200	9
4	5	3	200	9
4	9	4	200	12
5	6	5	350	3
5	9	6	400	9
6	7	7	500	5
6	10	8	250	13
7	8	9	250	5
7	11	10	300	9
8	2	11	500	9
9	10	12	550	10
9	13	13	200	9
10	11	14	400	6
11	2	15	300	9
11	3	16	300	8
12	6	17	200	7
12	8	18	300	14
13	3	19	200	11

. The free-flow time for regular vehicles (${𝓉}_a^1$) is calculated by applying a factor of 0.01 to the length, and for both types of trucks (${𝓉}_a^2,{𝓉}_a^3$), it is calculated by multiplying the free-flow time of regular vehicles by 1.2. The heterogeneous BPR functions are defined by ${\kappa }_a$, $l_a$, ${𝓉}_a^1$, ${𝓉}_a^2$, and ${𝓉}_a^3$ with the other parameters held consistent: ${\alpha }^1={\alpha }^2=0.3$, ${\alpha }^3=0.15$, and $\phi =4$. The OD demand consists of overloaded trucks ($q_{rp}^1\left(\mathbf{0}\right)$), non-overloaded trucks ($q_{rp}^2\left(\mathbf{0}\right)$), and regular vehicles ($q_{rp}^3\left(\mathbf{0}\right)$), with each accounting for 70%, 20%, and 10% of the total demand, respectively, as shown in

Table 2. OD demand for Nguyen–Dupuis network.

(veh/h)
Origin	Destination	Overloaded ${\mathit{q}}_{\mathit{r}\mathit{p}}^1\left(0\right)$	Non-Overloaded ${\mathit{q}}_{\mathit{r}\mathit{p}}^2\left(0\right)$	Regular ${\mathit{q}}_{\mathit{r}\mathit{p}}^3\left(0\right)$	Total
1	2	30	60	210	300
1	3	50	100	350	500
4	2	30	60	210	300
4	3	20	40	140	200

.

It is assumed that up to two WIMs can be installed on the network, i.e., $B=2$. The impact of WIM installation on travel costs and the resulting demand shifts is analyzed. The coefficients related to travel costs are defined as presented in

Table 3. Parameters to calculate travel cost.

${\mathit{\sigma }}^1,{\mathit{\sigma }}^2$ (USD/h/veh)	${\mathit{\sigma }}^3$ (USD/h/veh)	Fuel Price (USD/L)	Fuel Efficiency (km/L)			$\mathbf{Fuel}\mathbf{Efficiency}\mathbf{Cos}\mathbf{t}{\mathit{\gamma }}^{\mathit{i}}$ (USD/km)
			Regular	Non-Overloaded	Overloaded	Regular	Non-Overloaded	Overloaded
5	1	0.15	15	3.5	1.5	0.01	0.043	0.1

.

In the initial state, without any WIM installation ($W=\mathrm{0}$), traffic assignment is performed. As shown in

Table 4. Initial assignment result.

(veh/h)
Origin	Destination	Link	Regular	Non-Overload	Overload
1	5	1	341.79	97.66	48.83
1	12	2	218.21	62.34	31.17
4	5	3	170.00	48.57	24.29
4	9	4	180.00	51.43	25.71
5	6	5	377.44	107.84	53.92
5	9	6	134.36	38.39	19.19
6	7	7	377.44	107.84	53.92
6	10	8	104.90	29.97	14.99
7	8	9	202.23	57.78	28.89
7	11	10	175.20	50.06	25.03
8	2	11	315.54	90.15	45.08
9	10	12	132.15	37.76	18.88
9	13	13	182.21	52.06	26.03
10	11	14	237.05	67.73	33.86
11	2	15	104.46	29.85	14.92
11	3	16	307.79	87.94	43.97
12	6	17	104.90	29.97	14.99
12	8	18	113.30	32.37	16.19
13	3	19	182.21	52.06	26.03

, link 7, along with the connected link 5, exhibits the highest traffic volume of overloaded trucks. When a WIM is installed on link 7, overloaded trucks are unable to pass through and are required to take an alternative route. Under this situation, it appears reasonable to select link 7 as the initial candidate for WIM installation. However, detour traffic may result in additional congestion and adversely affect the pavement due to increased travel distances for overloaded trucks. Therefore, it is necessary to identify the optimal WIM installation links by considering both pavement management costs and travel costs for all possible WIM installation scenarios.

The utility of both non-overloaded and overloaded trucks is evaluated across all possible WIM installation cases. Overloaded trucks need to consider the additional benefits gained from overloading when determining their utility. These benefits, which are not affected by WIM installation, include the extra revenue generated by overloading. The benefit is calculated as USD 2.5 multiplied by the travel time of non-overloaded trucks. The decision on whether to switch demand is made by comparing non-overloaded and overloaded trucks for all cases. Since overloaded trucks carry more freight than non-overloaded trucks, it is assumed that one overloaded truck will be replaced by 1.5 non-overloaded trucks during the switch. For WIM installation cases and OD pairs where demand shifting occurs, the number of overloaded trucks is reduced by 50%, while the number of non-overloaded trucks increases by 1.5 times the reduced number of overloaded trucks. For each WIM installation case, a new traffic assignment is conducted for the adjusted demand, and pavement management costs are calculated using the parameters presented in

Table 5. Parameters to calculate pavement management cost.

${\mathit{M}}_{\mathit{a}}$ (USD)	${\mathit{b}}_{\mathrm{0}}$ (1/h)	${\mathit{b}}_1$ (1/(ESALs × hour))	${\mathit{s}}^{-}$ (m/km)	${\mathit{s}}^{+}$ (m/km)	$\mathbf{Vehicle}\mathbf{Specific}\mathbf{Group}\mathbf{Parameters}{\mathit{\lambda }}_{\mathit{i}}$
					Regular	Non-Overloaded	Overloaded
$3\times {10}^6$	0.040	$2\times {10}^{-5}$	4	1.5	0.0004	2.578	6.458

[20].

Figure 4 illustrates the process of deriving optimal WIM installation locations for the upper-level player in the first scenario by comparing the pavement management and travel costs of various WIM installation cases with the default case, marked as ‘x’, where no WIM is installed. As shown in

Table 6. Optimal points between travel cost difference and pavement management cost difference.

WIM Install Link	Pavement Management Cost Difference (USD/h)	Travel Cost Difference (USD/h)
5	−41,372 (−0.56%)	−46 (−0.34%)
3, 7, 14, 19 1 and 9, 1 and 12, 1 and 17, …, 11 and 19	−42,629 (−0.58%)	139 (+1.05%)

, Pareto-optimal WIM installation locations are identified for varying $\omega$. Among them, installing a WIM only on link 5 leads to a reduction in both pavement management cost and travel cost, yielding a positive impact on the overall network. The second Pareto point is associated with lower pavement management costs but higher travel disruption costs compared to the first Pareto solution. Given the simplicity of the Nguyen–Dupuis network considered, it includes multiple solutions for the placement of either a single WIM or double WIMs. While the reduction rate, at less than 1%, may not seem particularly significant, installing a WIM station is still economical when considering the prorated WIM cost, which is approximately USD 0.94 per hour per lane for a Single Load Cell WIM [21]. Despite the relatively low installation cost, the benefits from WIM—measured in terms of total system costs reduction—amount to approximately USD 40,000, calculated simply using $\omega =1$, resulting in a significantly high B/C ratio.

In the second scenario, a new constraint called the maximum allowable disruption level is introduced. The difference between the travel costs with WIM installation and the travel costs in the default case must fall within this constraint. This constraint varies depending on the government’s target congestion level for the network. For instance, if the target value is set to USD 100/h WIM installation locations must satisfy this constraint. As shown in Figure 5, applying this constraint excludes the second Pareto point identified in the first scenario, and the sole feasible optimal solution is found.

5. Discussion

This study examines the impact of WIM installation from a network-level perspective, incorporating various stakeholders, including the government, agencies, and drivers, through the framework of game theory. We propose a methodology that accounts for both the total travel costs of drivers and the pavement management costs associated with road maintenance. In this framework, the authorities responsible for installing WIM and the drivers who use the roads engage in a Stackelberg game, while drivers compete in a Nash game to minimize their individual travel costs. Consequently, this study provides a more realistic WIM installation strategy by incorporating the interactions among various stakeholders using game theory.

We assume demand shifts and traffic assignments are based on deterministic and homogeneous utility functions. However, in reality, each driver’s utility function should be defined in a stochastic and heterogeneous manner. Therefore, a stochastic approach that accounts for driver-specific variability would more accurately reflect real-world conditions. Additionally, since significantly damaged roads reduce fuel efficiency and driving safety for drivers, these costs should be integrated into driver costs. Finally, this study assumes that the pavement management agencies are affiliated with the government or local authorities (not fully independent), such as public corporations or government-affiliated organizations. However, the case where pavement management agencies exist as private corporations (completely independent) was not considered. Future research will aim to address this limitation.