Weigh-In-Motion Placement for Overloaded Truck Enforcement Considering Traffic Loadings and Disruptions

Abstract

Overloaded trucks directly contribute to road infrastructure deterioration and undermine safety, posing significant challenges to sustainability. This makes enforcement to reduce their numbers and impacts essential. Weigh-in-motion (WIM) systems use road-embedded sensors to measure truck weights and enforce regulations. However, WIM cannot be installed on all routes, and some overloaded truck drivers can detour to avoid them instead of giving up overloading if the detour penalty is still lower than the extra profit from overloading. This paper focuses on optimal WIM location planning for overloaded truck management, incorporating a demand shift and user equilibrium model based on the utility functions of overloaded and non-overloaded trucks. The presented framework includes an upper-level problem for WIM placement and a lower-level problem for demand shifts and traffic assignments among overloaded trucks, non-overloaded trucks, and light-duty vehicles for a given WIM placement. Particularly, at the upper level, the primary objective is to minimize the traffic loadings, i.e., the expected equivalent single-axle load–kilometers per unit time, with the secondary objective of minimizing the total traffic disruptions over the target network. Simulations and sensitivity analyses are conducted through a numerical example. Consequently, this study proposes an optimal WIM placement framework that considers drivers’ utility-based route choice and social costs such as ESAL and traffic congestion.

1. Introduction

Overloaded trucks are a significant cause of increased road maintenance and repair costs, directly leading to road infrastructure damage. Overloaded trucks cause structural distress to road pavements and reduce their service lifetime, with the loss cost due to overloaded trucks accounting for 60% of the total cost [1]. This is mainly due to overloaded axles creating significant non-uniformity in tire contact stress that increases plastic deformation of asphalt mixes [2,3]. The annual road damage cost due to overweight axles is approximately 20 to 30 million dollars [4]. Furthermore, the presence of overloaded trucks can increase pavement cost by more than 100% compared to the cost of the same vehicles with legal loads [5], and overloaded trucks from 0% to 20% can reduce the fatigue life of asphalt pavement in a range of 50%, while a 10% reduction in overloaded trucks may cause an increase in the service life of the pavement from 4 to 6 years [6]. Overloaded trucks also negatively affect bridges: an increase of 15% in vehicle gross weight can double bridge fatigue damage, thereby significantly reducing the structure’s lifetime [7].

To address these issues, it is essential to implement enforcement strategies that effectively reduce the number of overloaded trucks. Enforcement methods include stationary and mobile weigh stations. In stationary enforcement, inspection stations are established to monitor overloaded trucks using high-speed and low-speed axle load weighers. In mobile enforcement, portable axle load weighers are set up at selected inspection points.

Weigh-in-motion (WIM) is a stationary weigh station enforcement method that measures the weight and axle load of vehicles using sensors embedded in the road. WIM systems enforce regulations without requiring vehicles to slow down or stop, thus not disrupting traffic flow and even collecting various data. WIM can collect data such as truck type, axle load, and speed, which can be integrated with driver data [8]. Annual road damage costs were reduced by about USD 700,000 when WIM enforcement was conducted on the most damaged road sections in Montana [9]. This shows that WIM enforcement can effectively reduce road damage caused by overloaded trucks.

Despite these advantages, installing WIM systems on all roads is not feasible. Overloaded truck drivers often avoid inspection stations, with approximately 11–14% of overloaded trucks intentionally bypassing these stations [10]. This behavior highlights the need for strategic placement of WIM systems to monitor overloaded trucks effectively and encourage compliance with legal load limits.

The remainder of this paper consists of a literature review, methodology, numerical study, discussion, and conclusion.

2. Literature Review

Determining WIM installation locations is a type of flow capturing problem (FCP). The goal of the FCP is to locate facilities in order to intercept flow-based customers traveling between their origin and destination nodes [11]. The FCP is not only used for determining the locations of commercial facilities like convenience stores and gas stations but also the locations of vehicle inspection stations and traffic enforcement sites [11,12,13]. Additionally, one study designed the optimal locations of control stations using a mixed-integer programming model and an ant colony algorithm to maximize the truck traffic volume detected by enforcement vehicles [14]. However, this study failed to consider route variability and the severity of overloading. One of the main assumptions of FCP is that when facilities are located on predetermined routes, the flow is captured. Thus, drivers follow predetermined routes and are neutral to the facilities. However, in reality, drivers may evade inspection areas based on their experience.

To address the limitations of FCP, the evasive flow capturing problem (EFCP) considers that drivers can change their routes to avoid inspection facilities. One study examined the inspection facility location problem to minimize damage costs caused by overloaded trucks, using pre-generated routes and binary integer programming to solve the problem [8]. And other studies presented a multi-period stochastic EFCP [15], a bi-level program without pre-determined routes, using duality theory to convert it into a single-stage program and applying the branch-cut algorithm [16]. Additionally, one study modeled drivers’ route changes to avoid WIM sensors using bi-level programming, solving the problem by transforming it into a single-level problem through KKT conditions [17]. A pessimistic EFCP was also proposed, accounting for drivers’ limited rationality and generic cost functions [18].

There is a study that approached the WIM installation optimization problem by modeling the upper-level problem as determining WIM installation locations and the lower-level problem as drivers’ route choice, with the aim of eliminating possible evasion routes [19]. Another study determined the optimal locations and the number of WIMs in the upper-level problem and modeled all possible route evasions in the lower level, but it did not consider drivers’ demand shifts [20].

Some previous studies assume that overloaded truck drivers choose alternative routes within a deviation tolerance to avoid WIM locations and overloaded trucks continue overloading despite changing their routes (utility) [8,15,16,17,18,19,20]. For example, if a driver’s shortest route is now blocked by a WIM station, they might detour if the alternative route’s travel time or distance is within a preset deviation tolerance of the original path. However, in real situations, considering the additional travel cost penalty imposed for detours to avoid inspection points, drivers may compare the benefits of overloading with the benefits of non-overloading and choose not to overload. Thus, overloading demand can be converted to legal demand (non-overloading). The existing studies have the limitation of not adequately accounting for such realistic driver behavior.

To overcome these limitations, this paper derives the utility of overloaded and non-overloaded trucks according to the WIM locations and presents a demand shift and user equilibrium model that incorporates these utilities. When demand shifts due to differences in utility between the two types of trucks, link costs change, resulting in new utility values. This demand shift continues until the utility of non-overloaded and overloaded trucks equalizes, or no further demand shift occurs. Ultimately, using this equilibrium, the optimal locations for WIM installation are determined under a limited budget.

The contribution of this study is that it proposes a new approach to the WIM installation location planning by reflecting truck driver utility-based behavior. Through this, the study aims to effectively mitigate road damage and safety issues caused by overloaded trucks and contribute to the reduction of road maintenance costs.

3. Methodology

One of the major reasons for detecting and restricting overloaded trucks is to reduce their impact on roadway infrastructure by limiting their numbers. Thus, this study sets the primary objective to minimize the expected equivalent single-axle load–kilometers per unit time [ESALs-km/hr] across all vehicle groups, including overloaded trucks (Objective 1). At the same time, it is essential to ensure that the WIM systems do not significantly increase traffic congestion caused by detours of overloaded trucks. To address this, the secondary objective is set to minimize the total expected vehicle-hour-traveled (VHT) per unit time [veh-hr/hr], i.e., the total average number of vehicles on the target network, by all vehicle groups (Objective 2).

The planning problem is formulated as a bi-level optimization. The upper-level involves a bi-criteria problem for WIM location planning under a budget constraint, while the lower level addresses traffic distribution among three different vehicle groups: regular light-duty vehicles; non-overloaded trucks; and overloaded trucks. This bi-level structure facilitates a clear division of responsibilities and objectives among key stakeholders, ensuring both analytical simplicity and computational feasibility. The proposed framework is graphically summarized in Figure 1, with details provided hereafter.

3.1. Upper-Level WIM Planning Problem

A directed roadway network graph $G=(N,A)$ is considered, consisting of the sets of nodes and arcs, denoted by $N$ and $A$, respectively. The nodes and arcs are indexed by $n$ and $a$, respectively, and the length of arc $a$ is $l_a$ [km]. A unit WIM system can be installed on a candidate arc, and the set of candidate arcs is $A^{\prime }$. The decision is the set of the arcs with WIM is $X\left(\subseteq A\prime \right)$. Accordingly, binary variables $x_a$ can be defined, where it is one if a WIM is implemented on candidate arc $a$, and zero otherwise.

We consider pre-defined origin–destination (O-D) matrices before implementing WIM (i.e., when $X=0$ referring to an empty set), $q^{+}\left(0\right)$, $q^{-}\left(0\right)$, and $q^{car}\left(0\right)$ $\left(\in {\mathbb{R}}_{\ge 0}^{\left|N\right|\times \left|N\right|}\right)$ for overloaded trucks, non-overloaded trucks, and regular vehicles, respectively. For the O-D pair from node $n$ to node $n^{\prime }$, the demands for three classes are denoted by $q_{n{n}^{\prime }}^{+}\left(0\right)$, $q_{n{n}^{\prime }}^{-}\left(0\right)$, and $q_{n{n}^{\prime }}^{car}\left(0\right)$. After implementing WIM strategy $X\ne 0$, the resulting demands are $q^{+}\left(X\right)$, $q^{-}\left(X\right)$, and $q^{car}\left(X\right)$. This study focuses on the overloaded truck traffic response to WIM installation and assumes a constant $q^{car}$ independent of $X$ for analysis simplification. The set of the traffic flows on arc $a$ in [veh/hour] for three vehicle groups for given $X$ is $f_a\left(X\right)=\left\{f_a^{+}\left(X\right),f_a^{-}\left(X\right),f_a^{car}\left(X\right)\right\}$, and the set of link travel times on arc $a$ for three vehicle groups for given X is $t_a\left(X\right)=\left\{t_a^{+}\left(X\right),t_a^{-}\left(X\right),t_a^{car}\left(X\right)\right\}$. The set of the average ESAL per vehicle for three vehicle groups is {$F^{+}, F^{-}, F^{car}$}. Using this notation, the first and second objective functions are given as Equations (1) and (2), respectively.

The installation costs vary among candidate arcs, and the cost for candidate arc $a$ is $c_a$. The summation of WIM implementation cost must not exceed budget $B$, as described in Constraint 3. Lastly, Constraint 4 ensures that if at least one WIM is installed, the primary objective value is lower than that of the default case. The upper-level problem is summarized as the following mathematical programming:

$$\underset{X}{\min }\sum _a{(l}_a{f}_a^{+}\left(X\right)F^{+}+l_a{f}_a^{-}\left(X\right)F^{-}+l_a{f}_a^{car}\left(X\right)F^{car})(\mathrm{O}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e} 1)$$

(1)

$$\underset{x}{\min }\sum _{a\subset A}\left({t_a^{+}\left(X\right)f}_a^{+}\left(X\right)+{t_a^{-}\left(X\right)f}_a^{-}\left(X\right)+{t_a^{car}\left(X\right)f}_a^{car}\left(X\right)\right)(\mathrm{O}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e} 2)$$

(2)

s.t.

$$\sum _a{c}_a{x}_a\le B$$

(3)

$$\begin{gathered} \sum _a{(l}_a{f}_a^{+}\left(X\right)F^{+}+l_a{f}_a^{-}\left(X\right)F^{-}+l_a{f}_a^{car}\left(X\right)F^{car}) \\ <\sum _a{(l}_a{f}_a^{+}\left(0\right)F^{+}+l_a{f}_a^{-}\left(0\right)F^{-}+l_a{f}_a^{car}\left(0\right)F^{car}), if X\ne 0 \end{gathered}$$

(4)

3.2. Lower-Level Traffic Problem

The lower-level problem is to find $f\left(X\right)=\left\{f_a\left(X\right),\forall a\right\}$ for $X$ given from the upper-level problem. As generally assumed, non-overloaded truck drivers and regular light-duty vehicle drivers select their paths to minimize travel costs, including time and fuel costs [21]. Additionally, it is assumed that non-overloaded truck drivers will not transition to overloading after the implementation of WIM.

For the original overloaded truck drivers, i.e., the demand of $q^{+}\left(0\right)$, the overloading benefit is defined as the cost difference for an overloaded truck driver making a one-way trip between its origin and destination. This cost difference is calculated by comparing the cost on the lowest-cost path for an overloaded truck to the cost for the non-overloaded case on its respective lowest-cost path. Here, the lowest-cost path travel time and distance for overloaded trucks between $n$ to $n^{\prime }$ under $X=0$ are $t_{n{n}^{\prime }}^{+}{\left(0\right)}^{*}$ and $d_{n{n}^{\prime }}^{+}{\left(0\right)}^{*}$. Those for non-overloaded trucks and regular vehicles are $t_{n{n}^{\prime }}^{-}{\left(0\right)}^{*}$, $d_{n{n}^{\prime }}^{-}{\left(0\right)}^{*}$ and $t_{n{n}^{\prime }}^{car}{\left(0\right)}^{*}$, $d_{n{n}^{\prime }}^{car}{\left(0\right)}^{*}$. The time value of truck drivers is ${\beta }^{truck}$, that of regular vehicle drivers is ${\beta }^{car}$[USD/hour], and the fuel efficiency costs are ${\gamma }^{+}$, ${\gamma }^{-}$, ${\gamma }^{car}$ [USD/km] for overloaded, non-overloaded trucks, and regular vehicles, respectively, such that ${\gamma }^{+}\ge {\gamma }^{-}$, since overloaded trucks are heavier. In summary, overloading occurs because of the following:

$${\beta }^{truck}{t}_{n{n}^{\prime }}^{+}{\left(0\right)}^{*}+{\gamma }^{+}{d}_{n{n}^{\prime }}^{+}{\left(0\right)}^{*}-{\epsilon }_{n{n}^{\prime }}<{\beta }^{truck}{t}_{n{n}^{\prime }}^{-}{\left(0\right)}^{*}+{\gamma }^{-}{d}_{n{n}^{\prime }}^{-}{\left(0\right)}^{*}$$

(5)

where ${\epsilon }_{n{n}^{\prime }}$ is the extra income from overloading between $n$ and $n\prime$. The overloading benefit, denoted as $h_{n{n}^{\prime }}\left(0\right)$, equals the left-hand side (LHS) minus the right-hand side (RHS) of the above inequality.

The overloading penalty $\mathsf{∆}$ occurs when an overloaded truck passes a WIM. We assume that the overloading penalty is significantly higher than the overloading benefit, i.e., $∆\gg h_{n{n}^{\prime }}$ for all pairs of $n$ and $n^{\prime }$. The cost to travel arc $a$ with WIM for overloaded trucks is ${\beta }^{truck}{t}_a^{+}\left(X\right)+{\gamma }^{+}{l}_a+∆$, while that for non-overloaded trucks is ${\beta }^{truck}{t}_a^{-}\left(X\right)+{\gamma }^{-}{l}_a$, and for regular vehicles, it is ${\beta }^{car}{t}_a^{car}\left(X\right)+{\gamma }^{car}{l}_a$. These travel times can be directly determined by $f_a=\left\{f_a^{+},f_a^{-},f_a^{car}\right\}$, and they are eventually indirect functions of $X$ since $f_a$ is influenced by $X$ like $t_a^{+}\left(X\right)$, $t_a^{-}\left(X\right)$, and $t_a^{car}\left(X\right)$.

The Bureau of Public Roads (BPR) function is a widely used link cost function that derives link performance based on link capacity and demand [22]. In this study, the following heterogenous BPR functions are employed to estimate the travel times for trucks ($t_a^{+}\left(f_a\right)$ and $t_a^{-}\left(f_a\right)$) and regular vehicles ($t_a^{car}\left(f_a\right)$) on arc $a$, as functions of multi-group traffic $f_a$:

$$t_a^{+}\left(f_a\right)=t_a^{-}\left(f_a\right)=t_a^{0,truck}\left\{1+0.15{\left({\displaystyle \frac{f_a^{car}+\sigma \left(f_a^{+}+f_a^{-}\right)}{c_a}}\right)}^4\right\}$$

(6)

$$t_a^{car}\left(f_a\right)=t_a^{0,car}\left\{1+0.15{\left({\displaystyle \frac{f_a^{car}+\sigma \left(f_a^{+}+f_a^{-}\right)}{c_a}}\right)}^4\right\}$$

(7)

where $t_a^{0,truck}$ and $t_a^{0,car}$ are the free flow travel times of arc $a$ for trucks and regular vehicles, respectively; $c_a$ means the capacity of arc $a$ for mixed traffic; and $\sigma$ is a dimensionless weight coefficient to convert truck traffic to regular traffic. It is assumed that the free flow speed and the impact on the traffic environment are the same for both normal and overloaded trucks.

If the cost of overloading drivers on the new optimal path after implementing $X$ with path time and distance $t_{n{n}^{\prime }}^{+}{\left(x\right)}^{*}$ and $d_{n{n}^{\prime }}^{+}{\left(x\right)}^{*}$ is higher than the new optimal path when not overloading, the overloaded truck drivers will refuse to overload. Between $n$ and $n^{\prime }$, the number of reduced overloaded trucks $y_{n{n}^{\prime }}$ increases until overloading brings a benefit or until $y_{n{n}^{\prime }}$ becomes $q_{n{n}^{\prime }}^{+}\left(0\right)$ (i.e., all overloaded trucks have transitioned to normal trucks). Additionally, the conversion factor $\alpha$ is considered, representing the increase in the number of non-overloaded trucks when a single overloaded truck is reduced. This factor is assumed to be greater than 1. If the overloading reduction is denoted by $y=\left\{y_{n{n}^{\prime }},\forall n,n^{\prime }\right\}$, this non-refusal (keeping overloading) condition is given as follows:

$${\beta }^{truck}{t}_{n{n}^{\prime }}^{+}{\left(X,y\right)}^{*}+{\gamma }^{+}{d}_{n{n}^{\prime }}^{+}{\left(X,y\right)}^{*}-{\epsilon }_{n{n}^{\prime }}<{\beta }^{truck}{t}_{n{n}^{\prime }}^{-}{\left(X,y\right)}^{*}+{\gamma }^{-}{d}_{n{n}^{\prime }}^{-}{\left(X,y\right)}^{*}$$

(8)

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

According to the mathematical model for supernetworks with interdependent mixed traffic [23], the user equilibrium is found by solving the following assignment problem for given $X$ and $y$:

$$\begin{gathered} \underset{f|X,y}{\min }z\left(f\right|X,y)=\sum _{a\in A-X}{\int }_0^{f_a^{+}}\left[{\beta }^{truck}{t}_a^{+}\left(\omega ,f_a^{-},f_a^{car}\right)+{\gamma }^{+}{l}_a\right]d\omega \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} +\sum _{a\in X}{\int }_0^{f_a^{+}}\left[{\beta }^{truck}{t}_a^{+}\left(\omega ,f_a^{-},f_a^{car}\right)+{\gamma }^{+}{l}_a+∆\right]d\omega \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} +\sum _{a\in A}{\int }_0^{f_a^{-}}\left[{\beta }^{truck}{t}_a^{-}\left(f_a^{+},\omega ,f_a^{car}\right)+{\gamma }^{-}{l}_a\right]d\omega \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} +\sum _{a\in A}{\int }_0^{f_a^{car}}\left[{\beta }^{car}{t}_a^{car}\left(f_a^{+},f_a^{-},\omega \right)+{\gamma }^{car}{l}_a\right]d\omega \end{gathered}$$

(9)

s.t.

$$\begin{gathered} \sum _k{q}_{n{n}^{\prime },k}^{+}=q_{n{n}^{\prime }}^{+}\left(0\right)-y_{n{n}^{\prime }},\sum _k{q}_{n{n}^{\prime },k}^{-}=q_{n{n}^{\prime }}^{-}\left(0\right)+\alpha y_{n{n}^{\prime }}, \sum _k{q}_{n{n}^{\prime },k}^{car} \\ =q_{n{n}^{\prime }}^{car}\left(0\right), \forall n, n^{\prime } \end{gathered}$$

(10)

$$q_{n{n}^{\prime },k}^{+},q_{n{n}^{\prime },k}^{-},q_{n{n}^{\prime },k}^{car}\ge 0, \forall k,n, n^{\prime }$$

(11)

$$f_a^{+}=\sum _n\sum _{n^{\prime }}\sum _k{q}_{n{n}^{\prime },k}^{+}{\delta }_{a,n{n}^{\prime },k}^{+},$$

(12)

$$f_a^{-}=\sum _n\sum _{n^{\prime }}\sum _k{q}_{n{n}^{\prime },k}^{-}{\delta }_{a,n{n}^{\prime },k}^{-},$$

(13)

$$f_a^{car}=\sum _n\sum _{n^{\prime }}\sum _k{q}_{n{n}^{\prime },k}^{car}{\delta }_{a,n{n}^{\prime },k}^{car}$$

(14)

where $q_{n{n}^{\mathrm{\prime }},k}^{+}$, $q_{n{n}^{\mathrm{\prime }},k}^{-}$, and $q_{n{n}^{\mathrm{\prime }},k}^{car}$ are the flows on path $k$ between nodes $n$ and $n^{\mathrm{\prime }}$ for three vehicle groups; and ${\delta }_{a,n{n}^{\mathrm{\prime }},k}^{+}$, ${\delta }_{a,n{n}^{\mathrm{\prime }},k}^{-}$, and ${\delta }_{a,n{n}^{\mathrm{\prime }},k}^{car}$ are binary variable variables indicating if arc $a$ is included in path $k$ between nodes $n$ and $n^{\mathrm{\prime }}$. Equation (9) is to satisfy the Wardrop equilibrium conditions, and Constraints 10–14 ensure flow conservation and non-negativity. Conventional solution methodologies for user equilibrium (UE) can be utilized for this problem.

The above UE is not designed to find $y$ but is based on a given $y$. For a given $X$, an iterative algorithm is presented to find the best $y\left(X\right)$ in

Table 1. Numerical algorithm to find the reduced number of overloaded trucks for given WIM strategy.

$\mathbf{Input}: \mathit{X}$$, {\mathit{q}}^{+}\left(0\right)$$, {\mathit{q}}^{-}\left(0\right)$$, {\mathit{q}}^{\mathit{c}\mathit{a}\mathit{r}}\left(0\right)$, and the Other Parameters
Initialization: Set $y_{n{n}^{\prime }}\leftarrow 0$, $\forall n,n^{\prime }$. Step 1. Based on the current $y$, solve UE to find $f\left(y,X\right)$. Step 2. Numerically update $y_{n{n}^{\prime }}$ by increasing it with a step positively related to the overloading penalty, $-h_{n{n}^{\prime }}\left(x,y\right)\equiv \beta t_{n{n}^{\prime }}^{-}{\left(x,y\right)}^{*}+{\gamma }^{-}{d}_{n{n}^{\prime }}^{-}{\left(x,y\right)}^{*}-\left(\beta t_{n{n}^{\prime }}^{+}{\left(x,y\right)}^{*}+{\gamma }^{+}{d}_{n{n}^{\prime }}^{+}{\left(x,y\right)}^{*}\right)-{\epsilon }_{n{n}^{\prime }},$ $\forall n,n^{\prime }$. Step 3. If $y$ converges terminate. Otherwise, go to Step 1.
Output: $y\left(X\right)$ and $f\left(X\right)$

.

Using the output $y\left(X\right)$ and $f\left(X\right)$ of the algorithm for $X$ as the outcome of the lower-level problem, we numerically solve the combinatorial upper-level problem.

4. Numerical Study

4.1. Network Configuration

A numerical study is conducted using the Sioux Falls network, which includes 24 nodes and 76 links. The origin and destination nodes for overloaded and non-overloaded trucks are established by imitating the hub-and-spoke structure frequently used in real logistics industry. As illustrated in Figure 2, the origin nodes $n$ are set to 8, 10, 15, 19, and 22, while the destination node $n\prime$ are set to 1, and 13. For the default case with no WIM installation on any link, the hourly traffic demand of both overloaded and non-overloaded trucks, such as $q_{n{n}^{\prime }}^{+}\left(0\right)$, $q_{n{n}^{\prime }}^{-}\left(0\right)$, is detailed in

Table 2. Demand for trucks when WIM is not installed $(X=0)$.

(veh/hour)
$\mathbf{Origin} \left(\mathit{n}\right)$	$\mathbf{Destination} \left(\mathit{n}\mathit{\prime }\right)$
	1		13
	$\mathbf{\text{Non-Overloaded}} \left({\mathit{q}}_{\mathit{n}1}^{-}\left(0\right)\right)$	$\mathbf{Overloaded} \left({\mathit{q}}_{\mathit{n}1}^{+}\left(0\right)\right)$	$\mathbf{\text{Non-Overloaded}} \left({\mathit{q}}_{\mathit{n}13}^{-}\left(0\right)\right)$	$\mathbf{Overloaded} \left({\mathit{q}}_{\mathit{n}13}^{+}\left(0\right)\right)$
8	200	100	200	100
10	200	100	200	100
15	200	100	200	100
19	200	100	200	100
22	200	100	200	100

. The hourly traffic demand for regular vehicles $q_{n{n}^{\prime }}^{car}\left(0\right)$ on the network is detailed in

Table 3. Demand for regular vehicles $(q_{n{n}^{\prime }}^{car}\left(0\right))$.

(veh/hour)
	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24
1	0	50	50	250	100	150	250	400	250	650	250	100	250	150	250	250	200	50	150	150	50	200	150	50
2	50	0	50	100	50	200	100	200	100	300	100	50	150	50	50	200	100	0	50	50	0	50	0	0
3	50	50	0	100	50	150	50	100	50	150	150	100	50	50	50	100	50	0	0	0	0	50	50	0
4	250	100	100	0	250	200	200	350	350	600	700	300	300	250	250	400	250	50	100	150	100	200	250	100
5	100	50	50	250	0	100	100	250	400	500	250	100	100	50	100	250	100	0	50	50	50	100	50	0
6	150	200	150	200	100	0	200	400	200	400	200	100	100	50	100	450	250	50	100	150	50	100	50	50
7	250	100	50	200	100	200	0	500	300	950	250	350	200	100	250	700	500	100	200	250	100	250	100	50
8	400	200	100	350	250	400	500	0	400	800	400	300	300	200	300	1100	700	150	350	450	200	250	150	100
9	250	100	50	350	400	200	300	400	0	1400	700	300	300	300	450	700	450	100	200	300	150	350	250	100
10	650	300	150	600	500	400	950	800	1400	0	2000	1000	950	1050	2000	2200	1950	350	900	1250	600	1300	900	400
11	250	100	150	750	250	200	250	400	700	1950	0	700	500	800	700	700	500	50	200	300	200	550	650	300
12	100	50	100	300	100	100	350	300	300	1000	700	0	650	350	350	350	300	100	150	200	150	350	350	250
13	250	150	50	300	100	100	200	300	300	950	500	650	0	300	350	300	250	50	150	300	300	650	400	400
14	150	50	50	250	50	50	100	200	300	1050	800	350	300	0	650	350	350	50	150	250	200	600	550	200
15	250	50	50	250	100	100	250	300	500	2000	700	350	350	650	0	600	750	100	400	550	400	1300	500	200
16	250	200	100	400	250	450	700	1100	700	2200	700	350	300	350	600	0	1400	250	650	800	300	600	250	150
17	200	100	50	250	100	250	500	700	450	1950	500	300	250	350	750	1400	0	300	850	850	300	850	300	150
18	50	0	0	50	0	50	100	150	100	350	100	100	50	50	100	250	300	0	150	200	50	150	50	0
19	150	50	0	100	50	100	200	350	200	900	200	150	150	150	400	650	850	150	0	600	200	600	150	50
20	150	50	0	150	50	150	250	450	300	1250	300	250	300	250	550	800	850	200	600	0	600	1200	350	200
21	50	0	0	100	50	50	100	200	150	600	200	150	300	200	400	300	300	50	200	600	0	900	350	250
22	200	50	50	200	100	100	250	250	350	1300	550	350	650	600	1300	600	850	150	600	1200	900	0	1050	550
23	150	0	50	250	50	50	100	150	250	900	650	350	400	550	500	250	300	50	150	350	350	1050	0	350
24	50	0	0	100	0	50	50	100	100	400	300	250	350	200	200	150	150	0	50	200	250	550	350	0

.

The Sioux Falls capacity data, including the number of lanes and capacity per lane, is adopted from a previous study [24]. Lastly, each vehicle’s ESAL is roughly calculated using the given ESAL calculation formula [1]. A truck is considered to have a legal payload of 11 tons, a curb weight of 14 tons, 2 axles, and 6 tires. It is assumed that non-overloaded trucks carry a payload of 11 tons, while overloaded trucks carry a payload 1.5 times greater than that of non-overloaded trucks. Consequently, the ESAL values are set as follows: $F^{+}$ = 6.458, $F^{-}$ = 2.578, and $F^{car}$ = 0.0004.

4.2. Traffic Assignment and Demand Shift

The Frank–Wolfe algorithm is an iterative optimization method for solving nonlinear optimization problems and is commonly applied to the user equilibrium (UE) problem [25]. To incorporate the heterogeneous BPR functions and constraints discussed in this paper, the Frank–Wolfe algorithm is slightly modified. An initial traffic assignment is performed without WIM installation, and from the results, the links with the highest overloaded traffic flow are identified. Considering their layout to avoid selecting multiple links on a single path, WIM candidate locations $A^{\prime }$ are designated as $\{5, 19, 44, 74\}$.

Assuming a WIM installation budget of $B=$ USD 80,000, it is determined that a maximum of two WIMs can be installed considering WIM installation costs [26]. Consequently, the cost and demand shifts are then analyzed for scenarios involving one or two WIM installations. The value of time for drivers, used in travel cost calculations, is set to ${\beta }^{car}=1$, ${\beta }^{truck}=5$. Fuel costs are calculated based on the fuel efficiency and fuel price for regular vehicles, non-overloaded trucks, and overloaded trucks, with fuel efficiencies of 15, 3.5, and 2.5 [$\mathrm{k}\mathrm{m}/\mathrm{L}$], respectively, and a fuel price set to 0.15 [$\mathrm{U}\mathrm{S}\mathrm{D}/\mathrm{L}$]. Overloaded trucks experience increased engine strain from exceeding weight limits, which leads to higher fuel consumption during operation. Consequently, their fuel efficiency is lower compared to the standard 11-ton truck.

From this, the fuel cost parameters ${\gamma }^{car}, {\gamma }^{+}$, and ${\gamma }^{-}$ are computed. Based on the parameter values, travel costs without WIM ($X=0$) are calculated, as shown in

Table 4. Travel costs when WIM is not installed $(X=0)$.

(USD/veh)
		Travel Cost
Origin	Destination	Overloaded Trucks $({\mathit{\beta }}^{\mathit{t}\mathit{r}\mathit{u}\mathit{c}\mathit{k}}{\mathit{t}}_{\mathit{n}{\mathit{n}}^{\mathit{\prime }}}^{+}{\left(0\right)}^{\mathit{*}}+{\mathit{\gamma }}^{+}{\mathit{d}}_{\mathit{n}{\mathit{n}}^{\mathit{\prime }}}^{+}{\left(0\right)}^{\mathit{*}})$	Non-Overloaded Trucks $({\mathit{\beta }}^{\mathit{t}\mathit{r}\mathit{u}\mathit{c}\mathit{k}}{\mathit{t}}_{\mathit{n}{\mathit{n}}^{\mathit{\prime }}}^{-}{\left(0\right)}^{\mathit{*}}+{\mathit{\gamma }}^{-}{\mathit{d}}_{\mathit{n}{\mathit{n}}^{\mathit{\prime }}}^{-}{\left(0\right)}^{\mathit{*}})$	Regular Vehicles $({\mathit{\beta }}^{\mathit{c}\mathit{a}\mathit{r}}{\mathit{t}}_{\mathit{n}{\mathit{n}}^{\mathit{\prime }}}^{\mathit{c}\mathit{a}\mathit{r}}{\left(0\right)}^{\mathit{*}}+{\mathit{\gamma }}^{\mathit{c}\mathit{a}\mathit{r}}{\mathit{d}}_{\mathit{n}{\mathit{n}}^{\mathit{\prime }}}^{\mathit{c}\mathit{a}\mathit{r}}{\left(0\right)}^{\mathit{*}})$
8	1	1.81	1.59	0.30
	13	3.99	3.58	0.67
10	1	3.96	3.58	0.66
	13	4.39	4.15	0.73
15	1	6.14	5.74	1.02
	13	4.47	4.21	0.74
19	1	6.55	6.11	1.09
	13	4.88	4.58	0.81
22	1	5.00	4.64	0.83
	13	3.20	3.03	0.53

.

For overloaded trucks, the profit from overloading, ${\epsilon }_{n{n}^{\prime }}$, is derived by applying a certain conversion ratio (=0.1 [USD/km]) to the lowest-cost path length of non-overloaded trucks. Based on this, a utility comparison is conducted between overloading and choosing not to overload. If the utility without overloading is higher for some overloaded trucks, they are converted to non-overloaded trucks. Using the converted demand, traffic assignment is performed again, and new travel costs and utilities for overloaded and non-overloaded trucks are derived, resulting in further demand shifts. The conversion rate for overloaded trucks is set to 1.5 non-overloaded trucks per overloaded truck, reflecting the higher cargo load. This process is repeated until the utility for overloaded and non-overloaded trucks converges, or the difference between the current and previous number of non-overloaded trucks is less than 0.1 to stabilize the process and achieve convergence.

4.3. Impact of WIM Installation on Objective 1 and Objective 2 for All WIM Candidate Sets

The travel costs of overloaded trucks according to the WIM installation locations are shown in

Table 5. Travel costs by WIM installation candidates.

(USD/veh)
	Travel Cost of Overloaded Truck by Origin
WIM Location Link	8		10		15		19		22
	1	13	1	13	1	13	1	13	1	13
74	1.79	4.08	3.96	5.14	6.18	6.86	6.6	7.28	6.34	7.02
5	1.85	4.02	4.05	4.43	9.86	4.43	9.68	4.75	9.86	3.15
19	3.93	5.24	4.02	4.42	6.23	4.55	6.66	4.90	5.05	3.27
44	1.81	4.01	3.96	4.52	6.37	4.55	6.75	4.93	5.09	3.27
74 and 5	1.83	4.07	3.97	5.17	9.89	6.82	9.75	7.24	9.85	6.97
74 and 19	3.96	5.76	4.05	5.16	6.24	6.89	6.66	7.31	6.35	7.01
74 and 44	1.79	4.09	3.95	5.17	7.85	8.46	8.27	8.88	6.50	7.11
5 and 19	3.98	5.22	4.08	4.37	10.11	4.51	10.53	4.91	10.26	3.23
5 and 44	1.86	4.02	4.06	4.37	10.75	4.42	9.79	4.82	9.86	3.15
19 and 44	3.93	5.35	3.99	4.49	6.47	4.65	6.86	5.04	5.18	3.36

. The demand shifts and objective function values according to WIM installation locations are summarized in

Table 6. Objective 1 and Objective 2 values after WIM installation.

WIM Location Link	Number of Overloaded Trucks [veh/hour] $(\mathbf{Reduction} \% \mathbf{from} \mathit{X}=0)$	[Objective 1] ESAL Kilometers by All Vehicle Groups [ESALs-km/hr] $(\mathbf{Reduction} \% \mathbf{from} \mathit{X}=0)$	[Objective 2] Vehicle Hours Traveled (VHT) by All Vehicle Groups [veh-hr/hr]
			Overloaded	Non-Overloaded	Regular	Total VHT $(\mathbf{Reduction} \% \mathbf{from} \mathit{X}=0)$
Default ($X=0$)	1000 (-)	213,500 (-)	674	1343	79,360	81,376 (-)
74	702 (−30%)	207,565 (−3%)	501	1689	79,820	82,010 (0.78%)
5	702 (−30%)	195,850 (−8%)	412	1799	79,711	81,922 (0.67%)
19	901 (−10%)	209,295 (−2%)	688	1387	79,380	81,455 (0.10%)
44	1000 (0%)	214,664 (1%)	686	1333	79,227	81,245 (−0.16%)
74 and 5	404 (−60%)	186,203 (−13%)	206	2180	80,346	82,732 (1.67%)
74 and 19	603 (−40%)	207,327 (−3%)	512	1736	80,034	82,282 (1.11%)
74 and 44	702 (−30%)	215,273 (1%)	550	1702	80,131	82,383 (1.24%)
5 and 19	603 (−40%)	191,727 (−10%)	427	1856	79,962	82,245 (1.07%)
5 and 44	702 (−30%)	195,838 (−8%)	412	1801	79,749	81,962 (0.72%)
19 and 44	901 (−10%)	210,888 (−1%)	701	1378	79,429	81,508 (0.16%)

. Table 6 shows that changes in Objectives 1 and 2 range from −13 to 1% and from −0.16 to 1.67%, respectively. This indicates that the reduction in ESALs from installing WIMs outweighs the increase in traffic disruption. In particular, installing WIMs on links 5 and 74 reduces Objective 1 by 13%, a result of decreasing overloaded truck traffic by 60%, making this scenario the best option unless Objective 2 is excessively high.

The objective values for all the WIM installation candidates are shown in Figure 3, with Objective 1 on the x-axis and Objective 2 on the y-axis. The Pareto solutions [27], which improve Objective 1 while considering the trade-off between the two objectives, are “links 74 and 5,” “links 5 and 19,” “link 5,” and “link 19.” Although traffic disruptions (Objective 2) increase significantly for “links 74 and 5,” this candidate achieves the highest improvement in traffic loading reduction (Objective 1). The default case is also considered a Pareto solution.

Among the non-Pareto solutions, “link 44” and “links 74 and 44” lead to an increase in Objective 1, meaning that they do not meet Constraint 4. For “links 74 and 44,” even a slight reduction in the demand for overloaded trucks causes detours by remaining overloaded trucks, leading to more traffic loading and congestion. For “link 44,” the installed WIM does not convert any overloaded trucks into non-overloaded trucks and instead worsens traffic disruptions.

4.4. Sensitivity Analysis

The value of ${\epsilon }_{n{n}^{\prime }}$ is calculated by multiplying the lowest-cost path of non-overloaded trucks by the fixed conversion ratio (= 0.1 [USD/km]). However, in the real world, overloaded trucks can achieve different levels of profit depending on the degree of overloading and the type of goods transported. To explore the impact of different conversion ratios, sensitivity analysis is conducted.

In addition to the previously applied conversion ratio of 0.1 [USD/km], conversion ratios of 0.2 and 0.3 [USD/km] are also applied to calculate the demand shift and objective values. The Pareto set candidates for the different conversion ratio scenarios are shown in Figure 4. As the value of the conversion ratio increases from 0.1 to 0.2, implying greater profits from overloading, the Pareto set for the conversion ratio of 0.2 is dominated by that of the conversion ratio of 0.1, since fewer drivers give up overloading when higher profits are expected. Moreover, the default case was not included in the Pareto set for the case with the conversion ratio of 0.1, but the case with the conversion ratio of 0.2 includes it. This means that WIM installation can always worsen traffic congestion to reduce the ESALs-km/hr of all vehicle groups when the motivation for overloading is stronger, highlighting a clear tradeoff between Objective 1 and Objective 2.

When the conversion ratio is 0.3, the default case is the only Pareto solution, meaning that if overloading profit is too high, installing WIM at a limited location might not be enough to mitigate its impact and could even worsen the situation shown in Figure 5. In this case, no single overloaded truck gives up overloading due to the installed WIM(s); instead, they detour and, as a result, increase both objective values. In this scenario, the possibility of installing additional WIM systems under a higher budget is considered. Alternatively, other regulatory enforcement strategies may be employed alongside WIM installation, such as deploying non-expensive vision sensors at more locations to enhance overloading enforcement.

5. Discussion

Overloaded trucks directly cause and exacerbate road infrastructure damage, making it crucial to reduce their numbers through enforcement. WIM systems efficiently and effectively monitor overloaded trucks using sensors embedded in the road to measure their weight and axle load. This study proposes a decision-making framework to determine the optimal locations for WIM installation within the constraints of a limited budget. This framework is formulated as a bi-level problem, consisting of (i) an upper level with two objectives of minimizing traffic loading in ESALs (Objective 1) and minimizing traffic disruption (Objective 2); and (ii) a lower level focused on utility-based demand shifts and traffic distribution. Using the Sioux Falls network for numerical analysis, the study identifies optimal WIM locations based on this framework. Sensitivity analysis is conducted by varying the conversion ratio, which reflects the profit from overloaded trucks. The results demonstrate that strategically placed WIM systems can achieve the primary objective of minimizing traffic loading in ESALs.

This study represents one of the first attempts to develop a network-based understanding of WIM installations reflecting demand conversion. To ensure analytical simplicity, a deterministic model with homogeneous utility functions is adopted. However, real-world scenarios involve diverse and unpredictable driver behaviors, influenced by individual preferences, perceptions of enforcement risks, and other external factors. Addressing these uncertainties is crucial for improving the framework’s real-world applicability and robustness. Future research could incorporate stochastic user equilibrium (SUE) models to better capture variability, using stochastic elements such as Gumbel random variables in utility functions.

This study assumes that driver behavior stabilizes after WIM installations, focusing on the convergence state of truck drivers (overloaded and non-overloaded) and regular drivers. However, the perceived utility of overloaded truck drivers may undergo long-term changes due to the presence of WIM systems, potentially influencing their decision-making over time. Future research will aim to improve the understanding of these dynamics through driver surveys as stated preference data, as well as long-term historic data collected before and after actual WIM installations as revealed preference data to capture behavioral adjustments more accurately. Additionally, extending the framework to include additional sources of uncertainty, such as driver decision-making errors in route and mode choices, could offer deeper insights.

Lowering traffic loading can result in significant cost savings and greenhouse gas (GHG) reductions over several decades of planning horizons [28,29,30]. Integrating pavement maintenance, rehabilitation, and reconstruction (MR&R) processes into the WIM-related decision-making framework could provide more efficient and sustainable solutions in the context of pavement management systems (PMS). This integration facilitates the joint optimization of WIM placement and PMS strategies, addressing both enforcement goals and long-term infrastructure needs.

Lastly, while the Sioux Falls network provides a valuable testing ground as a toy network widely used in academic studies, the practical applicability of the proposed framework needs to be further validated using real-world networks with greater complexity. Specifically, in South Korea, roadway networks with potential WIM installation sites, such as the Saemangeum Northern Road and Muwang Road, are under investigation by the present authors’ research team. Current traffic volume and loadings, collected by loop detectors and sampled axle loads, are used to estimate O-D demands for multiple vehicle types and to calibrate the model parameters. Local governments provide preferred and available locations for WIM installation, which serve as candidate sites in the upper-level problem. Based on site-specific conditions, such as the lane number and pavement specifications, the expected WIM installation costs can be accurately determined.