Optimizing noise exposure in the Vehicle Routing Problem: A case study of last-mile freight deliveries in Stockholm

Siddharth Venkataraman 1

The Centre for ECO 2 Vehicle Design, The Marcus Wallenberg Laboratory for Sound and Vibration Research (MWL), KTH Royal Institute of Technology, SE-100 44, Stockholm, Sweden

Sacha Baclet The Centre for ECO 2 Vehicle Design, The Marcus Wallenberg Laboratory for Sound and Vibration Research (MWL), KTH Royal Institute of Technology, SE-100 44, Stockholm, Sweden

Dr. Romain Rumpler The Centre for ECO 2 Vehicle Design, The Marcus Wallenberg Laboratory for Sound and Vibration Research (MWL), KTH Royal Institute of Technology, SE-100 44, Stockholm, Sweden

ABSTRACT Improvements in noise mapping techniques and smart cities infrastructure have fostered the development of new ways to evaluate tra ffi c noise exposure. An example of one such outcome is the noise exposure sensitivity map, which quantifies the noise exposure potential of a road network as a function of the vehicle type, the prevailing background noise, and the population exposed. The potential for planning vehicle routing that is o ff ered by the above-mentioned map calls for solving the Vehicle Routing Problem (VRP) with a focus on optimising the noise exposure. A case study is chosen for applying the VRP, and it is taken from last-mile o ff -peak deliveries performed in Stockholm, Sweden in the context of the CIVITAS Eccentric project. The VRP is independently solved for the following objectives: distance travelled, driving time, and driving noise exposure potential. Also considered is a heterogeneous objective that is a combination of these factors. The impact of the objective function, on the resulting routes is presented.

1. INTRODUCTION

Freight transportation, although having benefits, also generates costs that fall on people in the vicinity of its operations [1]. This led to expanding the objectives of the classical vehicle routing

1 sidven@kth.se

a slaty. inter.noise 21-24 AUGUST SCOTTISH EVENT CAMPUS O ¥, ? GLASGOW

problem (VRP) to include metrics that can assess the potential for environmental pollution [2, 3]. Environmental noise has been regarded for decades as a cause of public health concern [4]. Its negative impact as an environmental stressor in Europe is increasing in dominance [5] with road tra ffi c noise considered as the main contributor [6]. In the context of road tra ffi c, road freight transport has a significant presence in the road network [7] and consequently shares in being responsible for noise pollution problem. Noise emissions is therefore a reasonable candidate to be considered in the VRP. The work presented here proposes a method to include noise pollution in the center of the vehicle routing problem, namely in the definition of the objective function. A prerequisite to apply this method is a road-network derived using a particular methodology for noise evaluation: receiver- centric noise exposure sensitivity mapping [8]. To test the methodology, an existing case study of last-mile o ff -peak deliveries is utilized to frame a vehicle routing problem. This case study took place in the City of Stockholm, Sweden, in the context of the CIVITAS Eccentric project [9]. Four objective functions are defined and used for optimizing the vehicle route. The optimized routes are then evaluated on the basis of the following metrics: distance travelled, driving time, and noise exposure potential.

2. METHODOLOGY

A classical single-echelon VRP is defined by locations of the depot and customers, and the costs incurred in travelling between each pair of locations. It also includes the supply at the depot, demand of the customers, and the capacity of the vehicles performing the delivery. A double-echelon VRP includes the presence of warehouses for the intermediate storage of goods between the depots and the customers. Depots and warehouses tend to be placed at the outskirts of urban areas, thereby reducing the scope of noise pollution when routing between them. Therefore, consideration of noise exposure is limited to the routing between customers and a depot or warehouse.

2.1. Input Data The primary type of input data is connected to the road network. The spatial limits of the road network is defined by the spatial spread of the nodes, i.e. , the locations of the depots / warehouses and the customers. Geographical data of a road network is obtained from OpenStreetMap (OSM) [10]. Two additional attributes of the road network need to be derived for evaluating the cost metrics presented in this approach. The first one is the average speed of a vehicle on each road segment. This requires GPS waypoint information from the vehicle types of interest, and road segment attributes, namely road category and maximum speed limit. The former is obtained from sources responsible for the vehicle service scheduling, e.g., logistics service providers. The latter two can be obtained from OSM provided the road network data is well maintained, otherwise from other GIS service providers like Google and HERE. The second additional attribute is the noise exposure potential of the road network. This is obtained from noise-exposure sensitivity maps using the approach presented in [8].

2.2. Cost metrics A cost metric is an attribute of a road segment that serves two purposes: it may be included in an objective function that is to be minimized, thereby exerting an influence in the route planning; it may also be used to evaluate a route obtained after optimizing an objective function, even if the cost metric is not used in this objective function. Two metrics that are commonly optimized in the VRP are: i) distance travelled, and ii) driving time. In the presented approach, a new metric is introduced to enable minimizing noise pollution: iii) noise exposure potential. The three cost metrics are defined and quantified for each road segment as

follows.

2.2.1. Distance Travelled

The distance travelled is defined as the length of the road segment. This quantity is an implicit feature of road networks when they are represented using a Geographic Information System (GIS) such as when obtained from OSM. The length of a road segment is estimated using a projected coordinate reference system to transform the global coordinates of the road segments onto a planar grid. The projection uses a UTM zone that is applicable to the spatial domain under consideration. It is to be noted that the altitude of the road network is not implicitly taken into consideration. For road networks across terrain with significant slope, the elevation can be incorporated through models such as Digital Terrain Models.

2.2.2. Driving Time

The driving time is defined as the time taken for a vehicle to traverse the road segment. This metric is dependent on the length of the road segment and the vehicle speed. Although the former is well- defined and deterministic, the latter is dependent on factors such as vehicle type, time of day (to account for varying tra ffi c conditions), weekday vs. weekend tra ffi c patterns, etc. These factors add to the uncertainty in accurately estimating the average driving speed. Therefore, a statistical approach is required to estimate the average vehicle speed, which in turn yields the average driving time. In this approach, the average driving time is estimated using GPS waypoints obtained from vehicles on the road network. Depending on the spatial and temporal distribution of these waypoints, the average speed can be estimated to an appropriate resolution. GPS waypoints from a particular vehicle type that are distributed across the entire road network and across 24 hours of the day allow for estimating the average speed for each road segment as a function of vehicle type and time of day. On the other hand, GPS waypoints that are sparsely distributed across a particular dimension require aggregation in order to obtain statistically relevant results. In the case where waypoints are sparsely distributed across the road network, a road category-based aggregation is performed. All road segments are grouped together according to their road category (e.g., motorway, primary, secondary, etc.). The average speed of each road category is evaluated and broadcast to all road segments that belong to it. Irrespective of the level of aggregation, the average speed v R c is defined as

P p ∈ P R c v p card ( P R c ) , (1)

v R c =

where v p is the driving speed in km.h − 1 from a GPS waypoint p , and P R c is the set of GPS waypoints corresponding to a set R c of road segments, card ( P R c ) is the size of the set P R c . When spatial aggregation is performed according to road category, a set R c contains all the road segments r belonging to a road category c . When spatial aggregation is not required, each road segment r is considered as a separate category c and is therefore the only member of a unique set R c . The average speed for a particular road segment v r is then defined as

v r = v R c | r ∈ R c . (2)

The driving time T r is defined as

T r = D r

v r , (3)

where D r is the length of road segment r .

A similar method of aggregation, estimation and broadcast is performed when waypoints are sparsely distributed either temporally or across di ff erent vehicle types. Consequently, by aggregating speeds over a dimension, it is assumed that driving speeds do not depend on this particular dimension. In the following case study, the GPS waypoints from the vehicle of interest were sparsely distributed both spatially and temporally. The resulting average speed for each road category includes the assumption that this speed is identical for all road segments within that category, and is independent of the time of day.

2.2.3. Noise Exposure Potential

A novel cost metric introduced in this approach is the noise exposure potential attributed to each road segment of a road network. The noise exposure potential, first introduced in [8], is a metric that quantifies the potential of noise exposure from a specific vehicle on the road network. This metric takes into account the expected increase in noise levels (“exceedance”) at receiver points placed on the façades of buildings due to the passing of the vehicle, compared to the existing background noise, as well as the number of citizens who would be a ff ected by this exceedance. The noise exposure potential considered in this work is the "linear impact indicator" which is directly proportional to the exceedance. N ′ r is the noise exposure potential of a particular road segment r per unit time and it is defined as N ′ r = 1 card ( I r )

X

X

j ∈ J i ∆ L + i , j · NP j , (4)

i ∈ I r

where ∆ L + i , j is the exceedance caused by a vehicle at position i at receiver position j , and NP j is the number of citizens associated with this receiver. J i is the set of receiver positions a ff ected by the vehicle at position i . I r is the set of vehicle positions on road segment r and card ( I r ) is the size of the set I r . It is to be noted that the exceedance ∆ L + i , j in Equation 4 is a function of the vehicle noise source strength and the prevailing background noise ( i.e. , time of day). The vehicle source strength is in turn a function of the vehicle source model ( i.e. , vehicle type) and the driving speed. The noise exposure potential N ′ r in Equation 5 is independent of time. In order to compare routes with di ff erent paths and driving times, it is necessary to scale the noise exposure potential with the time spent on each road segment. Following the definitions of common noise metrics such as the LA eq , a linear relation between magnitude of noise exposure and duration of exposure is assumed. The resulting noise exposure potential N r for a road segment r is defined as

N r = N ′ r ∗ T r , (5)

where T r is the driving time obtained from Equation 3. The frequency at which the delivery vehicles enters the road network is controlled by the delivery schedule and not by the vehicle driving time. Since the VRP assumes a constant delivery schedule, the noise exposure potential for a route is obtained by the summation of N r over all the road segments in the route, and the sum is not normalized by the overall driving time. The road network represented by its noise exposure potential is the "noise-exposure sensitivity map". The maps used for the following case study are presented in Figure 1. These maps compare the noise exposure potential from two vehicle types: diesel and hybrid. In the following case study, the map for vehicle type diesel (Figure 1a) will be utilized.

2.3. Objective functions The intended outcome of a VRP is controlled by the definition of the objective function. The choice of this function influences the solution to a VRP, since the solution is a set of vehicle routes that minimizes it. Therefore, the desired objectives ( e.g. , lesser fuel consumption, shorter delivery times, etc.) need to be represented by appropriate metrics within the objective function.

(c) Legend

(a) Vehicle type: Diesel

(b) Vehicle type:Hybrid

Figure 1: Noise-exposure sensitivity maps over a part of the City of Stockholm, Sweden, at 5:00 AM from two types of vehicles: a Diesel truck in (a) and a Hybrid truck in (b). A legend denoting the magnitude of N ′ r (Equation 4) is shown in (c).

Two types of objective functions are considered: homogeneous and composite. A homogeneous objective function contains only a single cost metric, making the cost metric the sole objective to minimize. A composite objective function contains a combination of two or more cost metrics, allowing for optimizing across di ff erent cost dimensions. For each objective function, the VRP identifies routes that satisfy the routing constraints while minimizing the overall cost that is defined by it. Defining a composite objective function is not straightforward because it involves combining metrics that may be physically unrelated ( e.g. , noise exposure and time). As a preliminary approach to study the relationship between such metrics, an objective function is defined as a normalized linear combination of these metrics. A composite objective function O c can be defined as

O c = X

m ∈ M A m ∗ m , (6)

where A m is the normalization constant for a cost metric m , and M is the set of cost metrics that are included in the composite objective function. The normalization constant A m for a cost metric m is defined such that A m ∗ m opt = 1, where m opt is the least possible value for that cost metric over all possible routes. m opt can be obtained by optimizing a homogeneous objective function containing that cost metric and evaluating the incurred cost.

2.4. Constraints The VRP, by definition, incorporates certain constraints on the routing possibilities. In the case of the classical VRP, each route should start and end at a depot / warehouse, each customer should be visited by only one route, and no customer should be ignored. In the case of a capacitated VRP, the vehicle’s transport capacity should not be exceeded by the cumulative demands of customers along a particular route. In addition to these fundamental constraints, it is possible to define supplementary constraints using the cost metrics to enforce requirements that are relevant to a particular routing problem. For example, the usage of battery-powered vehicles may require a constraint on the maximum length of a route. Similarly, the requirement to satisfy local government noise regulations may call for defining an upper limit to the noise cost. The definition of these supplementary constraints are specific to the routing problem and require careful choice and application of the cost metrics. In this contribution, supplementary constraints will be not be considered.

Noise Exposure Potential N' 35000 30000 25000 20000 15000 10000 '5000

2.5. Optimization and route evaluation The set of routes that minimize a chosen objective function can be obtained after applying optimization strategies at two levels of the problem. As a prerequisite, all the cost metrics that are present in the objective function should be defined for each road segment in the road network. This allows for quantifying the change each road segment has in the value of the objective function. The first level of optimization is to obtain the least change in the value of the objective function when travelling between any two stops. The change in the value of the objective function arising from a route consisting of multiple road segments is the cumulative change from its individual entities. This implies finding the shortest path between each pair of nodes, with the length of this path quantified by the objective function. The set of shortest paths is obtained using Dijkstra’s algorithm, and the change in the value of the objective function due to each path is simultaneously evaluated. The second level is an integer programming optimization problem. Given the change that the shortest paths have in the value of the objective function along with the VRP constraints, a suitable solver ( e.g. , IBM-CPLEX [11]) can be utilized for obtaining the routes that satisfy the VRP while minimizing the value of the objective function. This final optimization yields a set of routes, each with a sequence of customers in the order of the delivery route. After optimization, the set of routes can be evaluated using cost metrics defined for each road segment along these routes. This allows for optimizing a route over a particular dimension and evaluating the impact it has on another dimension.

3. CASE STUDY

The presented approach allowed for the planning of vehicle routes that consider the vehicle’s noise exposure potential. To illustrate an application of this approach, an existing case study of last-mile o ff -peak deliveries that took place in Urban Stockholm during the night [9] was revisited. The original purpose of this case study was to evaluate the extent of noise pollution arising from a delivery vehicle when following static predetermined routes. In this revisit, the potential for noise exposure was incorporated prior to the route planning. This allowed for comparing routes that were noise-optimized with those that were not. The following case study was an example of a single-echelon distribution system, with a depot located outside the city limits. There were a total of six customer locations distributed across the City of Stockholm as shown in Figure 2. The depot was located more than 15 km south-west of the nearest customer (in this case (6): Liljeholmen). All the routes connecting the depot to the customers passed through a common location which was associated to the entry / exit of a major motorway that connects to the inner-city road network. To reduce the spatial spread of this case study, this common location was considered as a "pseudo-depot". The routing between the depot and the pseudo-depot was not included in the VRP. The original case study involved a single delivery vehicle that performed three distinct routes that covered the six customers in sets of two customers per route. In this case study, the vehicle was required to visit all the six customers in a single route, essentially reducing the VRP to a Travelling Salesman Problem (TSP). Furthermore, supplementary constraints ( e.g. , maximum allowable Distance Travelled) were not imposed on the VRP. These simplifications were motivated by the fact that increasing the solution space would capture larger variations when optimizing di ff erent objective functions. They also allow for comparing di ff erent cost metrics solely based on their contribution to the objective function. Due to this di ff erence in vehicle routing requirements, it is to be noted that the real routing that was performed in the original context of this case study would not be applicable for a direct comparison with the optimal routing performed in this case study.

3.1. Cost metrics The following three cost metrics were evaluated for each road segment r on the road network:

Figure 2: A map of the City of Stockholm with locations of the customers (numbered 1 to 6 in the legend) and a pseudo-depot (numbered 0).

1. D istance Travelled – D r

2. Driving T ime – T r

3. N oise exposure potential – N r

Derivation of all three metrics required geographic information of the road network and this was obtained from OpenStreetMap. The road network was assumed to be an undirected graph, i.e. , the cost of travelling between two points on the road network was independent of the travel direction. The GIS data of the road network yielded the length of each road segment D r , shown in Figure 3a. The driving time was estimated using GPS waypoints from the vehicle that was used in the original context of this case study. The GPS waypoints were sparse in both temporal and spatial dimensions. Therefore, the average driving speed was calculated using Equation 1 wherein P R c included waypoints corresponding to all road segments belonging to a particular road category, and measured across 24 hours. These average speeds were then broadcast to all road segments using Equation 2, and is shown in Figure 3b. Equation 2 and Equation 3 were then used to calculate the average driving time T r for each road segment. The noise exposure potential N r was obtained as mentioned in Section 2.3. The intermediate attribute N ′ r is represented in the maps in Figure 1. For this case study, the configuration of a diesel vehicle driving at 5:00 AM was considered ( i.e. , Figure 1a).

3.2. Objective functions for the VRP Multiple instances of the VRP were performed for the same set of nodes in Figure 2, each with a distinct objective function. An objective function contained one or more of the following cost metrics: Distance travelled, Driving Time, and Noise exposure potential. The objective functions considered in this case study were:

1. D istance Travelled – D

2. Driving T ime – T

3. N oise exposure potential – N

8000.0. : Psuedo-depot Folkungagatan : Gotgatan Hornsgatan Slussen Sveavagen : Lillenolmen 7000.0, 6000.0, 5000.0, 4000.0, 3000.0, 2000.0, 1000.0,

(a) Distance Travelled – D r

(b) Average driving speed – v r

Figure 3: Maps representing the distance travelled and the average driving speed for each road segment of the road network.

4. Normalized linear combination of D , T and N – O c .

The first three functions were homogeneous and the resulting optimized routes would attain the minimum cumulative cost over their respective cost metrics. The forth function was composite and was defined as a normalized linear combination of the three cost metrics, as shown in Equation 6.

3.3. Estimation and evaluation of optimal routes For each of the four objective functions, the shortest path between each pair of nodes and the change it causes in the value of the objective function were determined. These costs along with the fundamental constraints of a VRP (in this case a TSP) were the input to the integer programming algorithm. This yielded an optimal route for each objective function. All the routes were then evaluated according to the three cost metrics defined in Section 3.1. The optimal routes and their evaluation are presented in Section 4.

SE EESES

4. RESULTS

The optimal routes obtained for the four objective functions defined in Section 3.2 are presented in this section. The four optimal routes in Figure 4 showed variations in two ways. The shortest path between two nodes was not necessarily identical for all the objective functions. For example, despite connecting the same pair of points (0): Pseudo-depot and (6): Liljeholmen (see Figure 2 for the location indexing), the path between these nodes is identical in Figure 4a and Figure 4d but di ff erent from the path in Figure 4c. Another di ff erence across the four optimal paths was the sequence for visiting the six customers. For example, a di ff erent customer was chosen as the first to be visited in each of the four routes, namely (3): Hornsgatan in Figure 4a, (5): Sveavägen in Figure 4b, (1): Folkungagatan in Figure 4c and (2): Götgatan in Figure 4d. The optimal routes in Figure 4 were also evaluated on the basis of the three cost metrics defined in Section 3.1. This allowed for a comparative evaluation and this is shown in Figure 5. The cost metrics in Figure 5 showed a comparable magnitude of variation across the four optimal routes. The maximum variation was 1 . 56 for cost metric D , and the minimum was 1 . 35 for cost metric T . The three routes obtained using the homogeneous objective functions ( D , T and N ) have the least

(a) Optimized Distance travelled

(b) Optimized Driving Time

(c) Optimized Noise Exposure Potential

(d) Optimized Composite Objective Function

Figure 4: Optimal paths evaluated using the the four objective functions in Section 3.2. The start and stop for a path between two nodes is denoted by an ’O’ and an ’X’, respectively. The color of a path represents its position in the order of visited nodes.

depot + #1 #15 #2 #29 #3 coer #43 #5 #5 #6 #6 > depot

Figure 5: Comparative evaluation of the four optimized routes (each denoted by a particular color in the bar plots), evaluated across the three cost metrics: (a) Distance Travelled, (b) Driving Time and (c) Noise Exposure Potential.

magnitude for their respective cost metrics in Figures 5a, 5b and 5c, respectively. This was expected given that in these cases, the objective function was identical to the cost metric. The cost metrics evaluated for the route optimized for the composite objective function (Figure 4d) were intermediate in magnitude to that of the other routes. Since noise was introduced as a new metric to optimize, the two objective functions that incorporate noise ( N and O c in Section 3.2) made it possible to test the consequences of attempting noise-sensitive mobility planning. This was tested by evaluating the relative change in the three cost metrics when a route not optimized for noise ( i.e., routes in Figures 4a and 4b) was converted to one that was optimized for noise ( i.e., routes in Figures 4c and 4d). The resulting change in the cost metrics is shown in Figure 6. Figure 6 highlighted the change in distance travelled and driving time when noise exposure potential was prioritized in the route-optimization. A complete switch to the route optimized only on N led to more than 40% increase in the distance travelled, and about 30% in driving time. This produced a benefit of about 20% reduction in the noise exposure potential. On the other hand, in the case of the route initially optimized only for D , making a switch to the route optimized using the composite objective function O c reduced noise exposure by 20% while increasing the distance travelled by 10% and the driving time by only 1%. In the case of the other route initially optimized for T , the switch reduced noise exposure potential by 7% at the expense of increasing distance travelled and driving time by only 2% and 3%, respectively.

5. CONCLUSION

The presented approach proposed a method to incorporate the e ff ects of noise pollution within the framework of the Vehicle Routing Problem (VRP). This was realized using previously defined noise- exposure sensitivity maps which quantify the noise exposure potential due to a vehicle on a road network. In order to run a preliminary analysis with this approach, a simple transportation problem was required. An existing case study of last-mile o ff -peak deliveries in Urban Stockholm, performed as part of the CIVITAS Eccentric project, was revisited to perform optimal route-planning with

(a) (b) (c) 45,_Distance Travelled - D Driving Time - T Noise Exposure Potential - N 16 30 in 4 25 2 - J30 10 3 3 Fs ~7° 3 2, ~ Ez & Qs & 220 =, 10 4 10 5 2 oO o o Dept —Topt Nope Oc. opt Dept Topt — Nope Or. opt Dept opt — Nope Oc. apt lm Route with optimum Distance Travelled - D Route with optimum Driving Time - T. lM Route with optimum Noise Exposure Potential - N Route with optimum for the Composite Objective Function - Oc

Figure 6: Percentage change in the cost metrics when a route without noise in the objective function (left: route optimized for D in Figure 4a; right: route optimized for T in Figure 4c) is converted to a route with noise (hatched bar: route optimized for N in Figure 4c; solid bar: route optimized for O c in Figure 4d)

objective functions that include the noise exposure potential of the delivery vehicle. Three cost metrics were defined to evaluate the road network and subsequent routes. They were distance travelled D , driving time T and noise exposure potential N . Four di ff erent objective functions were defined for optimizing the routes. The first three were related to the three cost metrics, and the fourth was a composite objective function that included all three cost metrics. Implementing the presented approach requires availability of data such as well-defined GIS data of road networks, average vehicle speeds on the road network and population statistics. In this case study, the lack of temporally and spatially rich data required making simplifying assumptions on the road network characteristics and consequently the quantification of noise exposure. This placed limitations on the quality of the resulting routes and their evaluation. Nonetheless, the methodology allows for handling rich data that captures important dynamics in the vehicle driving behavior and noise characteristics within a road network. Route optimization over the objective functions was performed and this was followed by evaluating the routes based on three cost metrics. The impact of noise exposure was quantified and compared between the di ff erent routes. The routes that were optimized only with noise in the objective function performed poorly when evaluated according to the distance travelled and driving time metrics. On the other hand, the composite objective function enabled reaching a compromise. It produced a reasonable reduction in noise exposure, while requiring only a marginal increase in the distance travelled and the driving time.

ACKNOWLEDGEMENTS

The authors would like to thank the Centre for ECO2 Vehicle Design, which is funded by the Swedish Innovation Agency Vinnova (Grant Number 2016-05195). Funding from the J. Gustaf Richert foundation (Grant Number 2021-00697) is also gratefully acknowledged.

Percentage change [%] fa) Current route: Dope route (b) Current route: Tope route Li ry ra aw 40 ‘aD or ww im Switch current route to Nope route mm Switch current route to 0... route

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