An Electric Vehicle Routing Problem with Charging Stations and Flexible Time Window

An Electric Vehicle Routing Problem with Charging Stations

and Flexible Time Window

Master’s Thesis, MSc Technology and Operations Management, Faculty of Economics and Business, University of Groningen,

The Netherlands 21st March 2016 Aoer Zhang Student number: S2572257 Email: a.zhang.1@student.rug.nl Supervisor/university:

Abstract

Table of Content

2.1ELECTRIC COMMERCIAL VEHICLE AND URBAN CONSOLIDATION CENTER ... 6

2.2VEHICLE ROUTING PROBLEM (VRP) ... 7

3. PROBLEM DESCRIPTION ... 10

4. MATHEMATICAL MODEL ... 14

5. NUMERICAL EXPERIMENTS ... 18

5.1DESIGN OF EXPERIMENTS ... 19

5.1.1 Parameter generation and description ... 19

5.1.2 Overview of problem instances ... 21

5.2RESULTS AND DISCUSSIONS ... 22

5.2.1 Computational time limit ... 22

5.2.2 The effect of allowing recharging in the city center and flexible time window 23 6. CONCLUSIONS ... 26

1. Introduction

City logistics is a branch of urban management systems, which emphasizes on solving the problems related to urban freight transportation (Qureshi, Taniguchi, & Yamada, 2009). The development of urban freight transportation not only contributes to the trading activities, but also incurs congestion and environmental nuisances to affect the quality of life, particularly in the city center (Crainic, Ricciardi, & Storchi, 2004; Anderson, Allen, & Browne, 2005). In order to deal with the urban freight transportation problems, European cities started to focus on examining the possibilities of employing electric commercial vehicles (ECV) to urban consolidation center (UCC) operations (Van Duin, Tavasszy, & Quak, 2013). The idea of UCC is the freights from outside the city are consolidated with the objective to bundle inner-city transportation activities, in order to split the distribution activities inside and outside the city (Van Rooijen & Quak, 2010). Meanwhile, the ECV is in a small size with zero NO2 emission and low CO2 emission, so that it could be viewed as well suited to the UCC delivery cycle (Van Rooijen & Quak, 2010; Daniela, et al., 2014; Iwan, Kijewska, & Kijewski, 2014). Accordingly, addressing the urban freight transportation field by integrating the concept of utilizing the ECV with UCC operations will make certain contributions to alleviating the urban freight transportation problems.

vehicles to service before or after customer’s earliest and latest time window bounds due to the penalty policy (Taş, et al., 2014). This means early arriving vehicles must wait or incur a penalty, and any late arrival is associated with penalty certainly within the flexible time window bounds. Conversely, customers require a restricted time window to be served in VRP with Hard Time Window (VRPHTW) issues.

Despite the prior studies were done in this field, there is no study jointly consider flexible time window with electric commercial vehicle routing problem (EVRP). When investigating in EVRP, hard time window constraints have drawbacks in case of real-life circumstance. For instance, the limitation of battery capacity and the speed of battery recharging may affect the vehicle travel time to meet the service constraint of customers (Feng & Figliozzi, 2013; Van Duin, et al., 2013). Meanwhile, the vehicle can only recharge in the depot will increase the vehicle total travel distance compare to conduct the recharge operations inside the city center. However, late arrival may be caused by electric vehicle recharging operations, so that customers have to wait for the service vehicles (Haghani & Jung, 2005; Bhusiri, et al., 2014). The hard time window constraints have the potential to strongly increase the number of vehicles, in order to fulfill all the customers with their required service time window (Schneider, Stenger, & Goeke, 2014). Whereas, the electric vehicle is concerned with high acquisition costs, which is still viewed as negatively affecting its operational competitiveness (Feng & Figliozzi, 2012; Melo, Baptista, & Costa, 2014). Therefore, Schneider, et al. (2014) advice to investigate the electric vehicle routing problem with considering soft time window in future study in order to avoid the extra high operation costs. At the same time, it can support transportation companies to determine the routing of electric vehicles in a more sophisticated scheme.

realistic insight of the model. In 2012, the municipality of Groningen asked Buck Consultants International to document the circumstance of transportation in the city center of Groningen. Hence, we design the numerical experiments that based on the certain facts and figures of Groningen city in the document. We aim at investigating the benefits of combining the possibilities of recharging during the service and allowance of customer’s flexible time window. The contributions of this paper are in twofold: firstly, it tries to enrich the existing literature of EVRP in urban freight transportation. Secondly, in practical, the developed mathematic model aims to support the transportation companies to make decisions on EVRP when customers allow serving within the flexible time window.

The remainder of this research paper is organized as following: Section 2 discusses the literature related to utilizing electric vehicles for UCC operations, and VRP. In Section 3, the problem description is discussed. In section 4, the mathematical model is presented. Section 5 shows the design of numerical experiments and discussion. The conclusion is drawn in Section 6.

2. Literature Review

In this section, the utilization of electric commercial vehicles (ECV) to urban consolidation center (UCC) operations is discussed. Then a review of previous literature concerning the relevant quantitative model of vehicle routing problem (VRP) takes place.

2.1 Electric Commercial Vehicle and Urban Consolidation Center

In the 1970s, the concept of urban consolidation center was first suggested to reduce the capacity of transportation vehicles in urban areas (McDermott, 1975). The term “urban consolidation center” also refers to “urban/ city distribution center” to describe UCC in the literature (Allen et al., 2012). Van Duin (1999) defines the initial concept of UCC as:

the distribution center to the city center and vice versa”. Van Rooijen & Quak (2010)

mention the main advantage of establishing a UCC near the city: large vehicles for long-haul transport stop outside the city may benefit the urban area for reducing the air pollution, traffic congestion, and safety risks.

In recent years, prior studies indicate that there are benefits to the operation costs applying in Dutch cities, by employing ECV with a UCC near the city center. This is because ECV only consumes electrical powered, which is usually consider as zero NO2 emissions, very low CO2 emissions and low noise level (Iwan, et al., 2014). Van Duin, Quak, & Muñuzuri (2010) conduct an experiment to investigate the possibility of establishing a UCC and use ECV for delivering in the city of The Hague. The results of the experiment show that 8% of vehicle-kilometers are reduced. Van Rooijen & Quak (2010) conduct another research in the city of Nijmegen, to evaluate the use of ECV with a new type of UCC, which is called Binnenstadservice.nl (‘Inner city service’) in their case. Their results present that a reduction of vehicle-kilometers in the city center is 32% at most, and the decrease of travel time in the city center is 25%. Later, Van Duin, et al. (2013) develop an EVRP model to consider using the ECV with UCC operations. Using the city of Amsterdam as a stage, a variety of scenarios are conducted to evaluate the model. The researchers find out that 19% of vehicle-kilometers are reduced in the scenarios, which means integrating electric vehicles into the UCC concept will improve the efficiency of urban freight transportation. Further, Browne, Allen, & Leonardi (2011) conduct a trial in the city of London, to evaluate the use of a UCC with ECV. The results show that CO2 emission of per parcel delivery is decreased by 54% in their experiments. As mentioned above, previous researches present the benefits of employing the ECV to UCC operations. Thus, it is interesting to investigate the ECV utilization in urban freight transpiration with more sophisticated factors, such as deploying recharging operations during service, and concerning the allowance of the flexible time window.

2.2 Vehicle Routing Problem (VRP)

(Braekers, Ramaekers, & Van Nieuwenhuyse, 2016). In recent years, more and more researchers started to explore the Electric Vehicle Routing Problem (EVRP) in order to address the green urban freight transportation issues. The potential benefits of employing ECV to urban freight transportation is discussed in Section 2.1.

Schneider, et al., (2014) indicate that researchers pay less attention to the alternative fuels or possibility of recharging when building the quantitative models of EVRP. To our knowledge, Recharging Vehicle Routing Problem (RVRP) is firstly investigated by Conrad & Figliozzi (2011), which allows ECV to recharge during transportation service. Both battery range and energy consumption are taken into account in the research problem statement. However, the researchers assume the recharge operations are executed with a fixed duration, and the charging stations are located in every customer’s locations. Erdoğan & Miller-Hooks (2012) formulate a Green Vehicle Problem (GVRP) to support the conventional fuel vehicle with limited vehicle driving range due to the restriction of the fuel tank. A set of alternative fueling stations (AFSs) and the opportunities of visiting AFSs are introduced, for the purpose of allowing the vehicle being refueled. Yang & Sun (2015) focus on swapping the battery instead of recharging when the ECV arrive at the battery swap locations. This study aims at determining the station location as well as EVRP under battery driving range limitation. However, neither vehicle capacity limitation nor customer time window restriction is addressed in aforementioned studies.

different charging stations, due to the costs vary with different recharging technology. The researchers aim to minimize the total recharging costs and determine the amount of energy recharged in different charging stations. Later extension work is done by Hiermann et al. (2016). Their model aims at minimizing not only the transportation costs but also the electric vehicle acquisition costs. However, none of the aforementioned studies address the EVRP incorporating with the soft time window allowance.

On the other hand, numerical studies have been done regarding the extension of VRP with investigating in the soft time window (VRPSTW), although the mathematical models are given different names (Taillard et al., 1997; Qureshi, et al., 2009; Figliozzi, 2012). The reason for addressing soft time window is that it results to lower the routing costs via reducing the number of vehicles to use as well as decreasing the travel distance (Chiang & Russell, 2004). More recently, Bhusiri, et al. (2014) consider the Vehicle Routing Problem with Soft Time Window (VRPSTW). Early arrival penalties and late arrival penalties are introduced to their problem. That means vehicles arrive earlier than the customers’ required time window are associated with penalties instead of waiting for the start of service. Similar work is done by Taş, et al. (2014), the researchers concern a tolerance is given to the vehicles that allow deviations occur in customer time windows. The concept is presented as Vehicle Routing Problem with Flexible Time Windows (VRPFlexTW). A penalty for the arrival between flexible earliest allowable time for service and earliest allowable time for service is formulated in the model, and this also applies to the late arrival situation. The difference between VRPSTW and VRPFlexTW is that where the relaxation of the time window is unbounded in the former case (such as infinite flexible bounds), whereas relaxation of the time window is bounded in the later case. In VRPFlexTW, the researchers propose both the lower and upper bounds of the relaxed time window. Both of the studies conclude that relaxing the time window enables savings in the transportation costs, decreasing the time on waiting at the customer location, and reducing the number of vehicles for servicing.

allowance of the flexible time window. In fact, there is no prior research considering flexible time window together with EVRP. Therefore, the goal of our study is to fill this gap and shed a light on studies in the future of EVRP incorporating with the flexible time window. In this paper, we integrate the EVRP with vehicle capacity, battery range, dependent recharging duration, the possibility of recharging inside the city center, and allowance of the flexible time window of customers. In addition, the study of Erdoğan & Miller-Hooks (2012) provides us the techniques on the model building when generating the set of recharging stations and the opportunities of vehicle recharging. Based on the reviews of the extension of related VRP in current status, prior studies give us certain hints to have a deeper exploration of the problems in our study. In Section 3, the problem description of what we are addressing is discussed.

3. Problem Description

However, the recharging operations take a considerable amount of time, which may inevitably affect the route planning in order to serve all the customers within their required time window. In this situation, neither reducing the number of customers being serve by the same vehicle nor investing in more ECV is a profitable solution. Consequently, we introduce the concept of flexible time window of customers based on Taş, et al. (2014), as to relax the customer’s time window restriction and provide the possibility of recharging operations on route planning simultaneously. The researchers indicate that flexible time window takes a fixed relaxation of the time window constraints into account, and it has the potential to lower the routing costs resulted from fewer vehicles. The flexibility of time window is shown in Figure 3.1. The penalty will be charged when service starts between flexible earliest allowable time and earliest allowable time at each customer. This regulation also applies to the late service situation.

Figure 3.1 Possible arrivals and their corresponding penalty cases at any customer 𝑖 (Taş, et al., 2014).

window, which are named as nRnF, RnF, nRF, and RF. Prior studies indicate that comparing to nRnF, vehicle travel distance and total transportation costs are reduced in RnF situation. On the other hand, previously research also indicates that compared to nRnF, the number of vehicle used to serve the customers and total transportation costs are decreased in nRF situation. Therefore, we suppose a larger benefit can be achieved in RF situation compared to the two aforementioned situations. Our major objective is to investigate whether the combination of allowing recharging during service and implementing customer’s flexible time window for urban freight transportation will decrease the vehicle travel distance, reduce the number of vehicle used for delivery, and lower the total transportation costs. As mentioned in Introduction, we use the city of Groningen as a stage to perform the experiment and test the model. In practice, we aim to support the transportation companies to conduct eclectic vehicle routing planning by developing the mathematical model in our study.

Figure 3.2 Problem description

In this paper, the model of electric vehicle routing problem with flexible time window (EVRPFlexTW) will be presented. In order to clarify the problem in our study, the following assumptions are discussed.

1. Starting and ending nodes: the urban freight transportation will be initiated from the UCC, which is proposed to serve all or part of an urban area (Allen et al., 2012).

There is only one UCC operating near the city center in this study, and every delivery starts and ends at the UCC.

2. Customer visits: customers are the shops and restaurants located in the inner-city. Every customer has to be served and every customer is served exactly once, by exactly one ECV. It means the demand of each customer is satisfied by a delivery from only one ECV per day.

3. UCC operations: we assume that packages are ready to be transported in the UCC at the beginning of the day.

4. Consumption of transportation vehicle: the ECV use electricity power for the operations. The consumption of battery capacity of ECV depends on the distance of transportation.

5. Penalty cost for waiting: there is no penalty cost incurred for waiting in the EVRPFlexTW. This enables ECV to wait at customer locations although they arrive within the flexible time windows, in order to generate cost-efficient routes (Taş, et al., 2014).

Furthermore, a mathematical model needs to be simpler than the system it represents, which enables the analyst to predict the effect of changes to the system (Maria, 1997). Thus, even though the situation could be changed in real-world, we make following assumptions in order to simplify the model. We also explain the potential differences in reality circumstances.

6. Transportation vehicle: the activities that involved in the urban freight transportation are performed by ECV in this paper. All customers are served by a set of homogeneous electric vehicles. In real-world, transportation companies could invest in multiple types of ECV to satisfy different customer demands.

the geographic platform contains different grades that may influence the vehicle operating speed (Figliozzi, 2011).

8. Recharging operations: We assume a linear recharge in terms of recharging operations, although the recharging time may increase for the last 10% - 20% of battery capacity in reality (Schneider, et al., 2014). Meanwhile, we assume the ECV should wait at the recharging station till the battery meets its maximum capacity before continuing its tour, although in real-world the ECV could leave the charging station once the battery capacity can cover rest of the routes. Moreover, we also assume each charging station can be only used by one ECV simultaneously.

In the next section, a mathematical model which meet all the conditions mentioned above is presented.

4. Mathematical Model

In this section, we briefly describe the model and present the model formulation. We formulate the EVRPFlexTW as mixed-integer linear programming (MILP). The model is built based on the previous research of Schneider, et al. (2014) and Taş, et al. (2014), which including the recharging operational opportunities, and flexible time window allowance.

Let set 𝑉 = 1, 2, … , 𝑛 denotes the set of customers, and the nodes 0 and n + 1 denote the depot which starting and ending nodes of each route, respectively the urban consolidation center in this case. We have set 𝐹 = 𝑛 + 2, 𝑛 + 3, … , 𝑛 + 𝑓 as a set of charging stations. The dummy nodes set 𝐹/ _{= 𝑛 + 2 + 𝑓, 𝑛 + 3 + 𝑓, … , 𝑛 + 𝑓 + 𝑓}/ represents multiple times to permit multiple potential visits to each node in the set of charging stations 𝐹. In another word, the number of dummy nodes is associating with each charging station, which denotes the number of times can be visited. Let set 𝑉_0,123/ ₌ 0 ∪ 𝑉 ∪ 𝑛 + 1 ∪ 𝐹 ∪ 𝐹/_{, including nodes of depot, customer, and charging stations and} the associated potential visits. We have 𝐺 = 𝑉_0,123/ _{, 𝐴 as a graph with the set of nodes} where 𝑉_0,123/ _{= 0, 1, … , 𝑛 + 𝑓 + 𝑓}/ _{, and arcs 𝐴 = 𝑖, 𝑗 | 𝑖, 𝑗 ∈ 𝑉}

corresponds to the edges connecting nodes. Here, we use the research of Erdoğan & Miller-Hooks (2012) to support the technique of setting the dummy nodes.

A set 𝐾 = 1, 2, … , 𝑘 denotes homogeneous electric vehicles, which with vehicle capacity 𝐶 and battery capacity 𝑄 for each vehicle 𝑘 . We also have 𝑔 as the recharging time rate and 𝑐 as the charge consumption rate for each vehicle 𝑘. Variable 𝑦_CD indicate the remaining battery capacity of vehicle 𝑘 on arriving at node 𝑖. We define 𝑑CF and 𝑡CF as the travel distance and travel time with each arc 𝑖, 𝑗 ∈ 𝐴. Note that the service time at customer node 𝑖: 𝑠_C, is not included in 𝑡_CF. Moreover, cost 𝐶_I incurs for vehicle 𝑘 associates with each unit of traveling distance 𝑑_CF, and a fixed cost 𝐶_J incurs for using the vehicle.

We have a positive demand 𝑞_C for each customer 𝑖 ∈ 𝑉. For each customer 𝑖, there is a time window 𝑙_C, 𝑢_C , and fractions 𝑝_CO _{and 𝑝}

CP to set the maximum allowed violations, which is leading to the flexible time window. A flexible time window 𝑙_C/_{, 𝑢}

C

/ _{is defined} in regard to the length of the original time window for each customer 𝑖. In this study, 𝑙_C/ _{= 𝑙}

C − 𝑝CO 𝑢C − 𝑙C , and 𝑢C/ = 𝑢C+ 𝑝CP 𝑢C − 𝑙C . The technique of building the flexible time window is depending on the research of Taş, et al. (2014). For the time window violations, a penalty cost 𝐶_R incurs for one unit of earliness when serving a customer within 𝑙_C/_{, 𝑙}

C , and a penalty cost 𝐶S incurs for one unit of delay when serving a customer within 𝑢_C, 𝑢_C/ _{. Variable 𝜏}

CD specify the time on arrival at node 𝑖 of vehicle 𝑘, and variable 𝑒_CD _{and ℎ}

C

D _{indicate how many units of earliness and delay of vehicle 𝑘} when arriving on a customer 𝑖.

The sets, parameters, and variables of the model are summarized in below.

Sets

𝟎, 𝒏 + 𝟏 Depot instances

𝑭 Set of charging stations, indexed 𝑛 + 2, 𝑛 + 3, … , 𝑛 + 𝑓

𝑭/ Set of dummy nodes of potential visits to charging stations, indexed 𝑛 + 2 + 𝑓, 𝑛 + 3 + 𝑓, … , 𝑛 + 𝑓 + 𝑓/

𝑭_𝟎/ Set of charging stations, and visits to charging stations, including depot instance 0, 𝐹₀/ _{= 𝐹 ∪ 𝐹}/ _{∪ 0}

𝑽 Set of customers, indexed 1, 2, … , 𝑛

𝑽/ Set of customers, charging stations, and visits to charging stations, 𝑉/ = 𝑉 ∪ 𝐹 ∪ 𝐹/

𝑽_𝟎/ Set of customers, charging stations, and visits to charging stations, including depot instance 0, 𝑉₀/ _{= 𝑉}/_{∪ 0}

𝑽_𝒏2𝟏/ Set of customers, charging stations, and visits to charging stations, including depot instance 𝑛 + 1, 𝑉₁₂₃/ _{= 𝑉}/_{∪ 𝑛 + 1}

𝑽_{𝟎,𝒏2𝟏}/ Set of customers, charging stations, and visits to charging stations, including depot instance 0 and 𝑛 + 1, 𝑉_0,123/ _{= 𝑉}/_{∪ 0 ∪ 𝑛 + 1}

𝑲 Set of electric vehicles, indexed 1, 2, … , 𝑘

Parameters

𝑪_𝒕 Costs for one unit of traveling distance 𝑪_𝒇 Fixed costs for using the vehicle

𝑪_𝒆 Penalty costs for each unit of early servicing 𝑪_𝒅 Penalty costs for each unit of late servicing 𝒅_𝒊𝒋 Traveling distance for arc 𝑖, 𝑗

𝒒𝒊 Positive demand for customer 𝑖

𝑪 Vehicle capacity

𝑸 Battery capacity

𝒈 Recharging time rate

𝒄 Charge consumption rate

𝒕_𝒊𝒋 Traveling time for arc 𝑖, 𝑗 𝒔_𝒊 Service time for customer 𝑖

𝒍_𝒊 Earliest allowable time for service at node 𝑖 𝒖_𝒊 Latest allowable time for service at node 𝑖

𝒍_𝒊/ _{Flexible earliest allowable time for service at node 𝑖} 𝒖_𝒊/ _{Flexible latest allowable time for service at node 𝑖}

Variables

𝒙_𝒊𝒋𝒌 Binary variable, equals to 1 if electric vehicle 𝑘 travels for arc 𝑖, 𝑗 , otherwise 0

𝒚_𝒊𝒌 _{Remaining battery capacity of electric vehicle 𝑘 on arrival at node 𝑖}

𝝉_𝒊𝒌 _{Time of electric vehicle 𝑘 on arrival at node 𝑖}

𝒆_𝒋𝒌 _{Earliness of electric vehicle 𝑘 starts service at node 𝑗} 𝒉_𝒋𝒌 _{Delay of electric vehicle 𝑘 starts service at node 𝑗}

The mathematical model for the EVRPFlexTW is formulated as follows:

Subject to: 𝑥_CFD D∈u F∈vwxyz ,C{F = 1, ∀ 𝑖 ∈ 𝑉, (2) 𝑥_CFD D∈u F∈v_wxyz _,C{F ≤ 1, ∀ 𝑖 ∈ 𝐹/_{, (3)} 𝑥_0FD F∈vwxyz = 1, ∀ 𝑘 ∈ 𝐾, (4) 𝑥_C0D C∈v_|z = 1, ∀ 𝑘 ∈ 𝐾, (5) 𝑥_FCD _{− 𝑥} CFD C∈v_|z_,C{F C∈v_wxyz _,C{F = 0, ∀ 𝑗 ∈ 𝑉/_{, ∀ 𝑘 ∈ 𝐾, (6)} 𝑞_C C∈v 𝑥_CFD F∈v_{|,wxy,…†‡}z ≤ 𝐶, ∀ 𝑘 ∈ 𝐾, (7) 𝜏_CD_{+ 𝑡} CF+ 𝑠C 𝑥CFD − 𝑢0/ 1 − 𝑥CFD ≤ 𝜏FD, ∀ 𝑖 ∈ 𝑉0, ∀𝑗 ∈ 𝑉123/ , 𝑖 ≠ 𝑗, ∀ 𝑘 ∈ 𝐾, (8) 𝜏_CD_{+ 𝑡} CF𝑥CFD + 𝑔 𝑄 − 𝑦CD − 𝑢0/ + 𝑔𝑄 1 − 𝑥CFD ≤ 𝜏FD, ∀ 𝑖 ∈ 𝐹/_{, ∀𝑗 ∈ 𝑉} 123/ , 𝑖 ≠ 𝑗, ∀ 𝑘 ∈ 𝐾, (9) 𝑙_C/ _{≤ 𝜏} CD ≤ 𝑢C/, ∀ 𝑖 ∈ 𝑉0,123/ , ∀ 𝑘 ∈ 𝐾, (10) 𝑒_CD ≥ 𝑙C − 𝜏CD, ∀ 𝑖 ∈ 𝑉0,123/ , ∀ 𝑘 ∈ 𝐾, (11) ℎ_CD_{≥ 𝜏} CD− 𝑢C, ∀ 𝑖 ∈ 𝑉0,123/ , ∀ 𝑘 ∈ 𝐾, (12) 0 ≤ 𝑦_FD_{≤ 𝑦} CD− 𝑐 ∙ 𝑑CF 𝑥CFD + 𝑄 1 − 𝑥CFD , ∀𝑖 ∈ 𝑉, ∀ 𝑗 ∈ 𝑉123/ , 𝑖 ≠ 𝑗, ∀ 𝑘 ∈ 𝐾, (13) 0 ≤ 𝑦_FD_{≤ 𝑄 − 𝑐 ∙ 𝑑} CF 𝑥CFD, ∀𝑖 ∈ 𝐹0/, ∀ 𝑗 ∈ 𝑉123/ , 𝑖 ≠ 𝑗, ∀ 𝑘 ∈ 𝐾, (14) 𝑦₀D_{＝ 𝑄, ∀ 𝑘 ∈ 𝐾, (15)} 𝑒_CD _{≥ 0, ∀ 𝑖 ∈ 𝑉} 0,123/ , ∀ 𝑘 ∈ 𝐾, (16) ℎ_CD_{≥ 0, ∀ 𝑖 ∈ 𝑉} 0,123/ , ∀ 𝑘 ∈ 𝐾, (17) 𝑥_CFD ∈ 0, 1 , ∀ 𝑖 ∈ 𝑉₀/_{, ∀ 𝑗 ∈ 𝑉} 123/ , 𝑖 ≠ 𝑗, ∀ 𝑘 ∈ 𝐾. (18)

charging station and each of its associated dummy nodes is visited at most once by one vehicle. Constraints (4) indicate each vehicle route starts at the depot. Constraints (5) indicate each vehicle route ends at the depot. Constraints (6) guarantee the flow conservation that the number of incoming arcs is equal to the number of outgoing arcs. Constraints (7) ensure the vehicle capacity cannot be exceeded. Constraints (8) guarantee the time feasibility for arcs leaving customers and the depot. Constraints (9) guarantee the time feasibility for arcs leaving charging stations and the associated dummy nodes. 𝑔 𝑄 − 𝑦_CD _{indicate the required duration for recharging operations of a vehicle.} Constraints (10) ensure the starting time of service takes place at each customer within the customer’s flexible time window. Constraints (11) represent the relation ship between earliness and the beginning of service, and constraints (12) represent the relationship between delay and the beginning of service. Constraints (13) restrict the current battery capacity of the vehicle is reduced when coming from a customer node 𝑖. The reduction is considering with the distance between the node 𝑗 and its predecessor node 𝑖, and the vehicle’s charge consumption rate. The node 𝑗 can be a depot, a customer, or a charging station and its associated dummy node. Constraints (14) ensure the current battery capacity of the vehicle is reduced when coming from a charging station or its associated dummy visits node 𝑖. Constraints (15) ensure the battery capacity remains in the vehicle is full charged at the starting depot node 0. Constraints (16) and (17) guarantee the earliness and delay are with non-negative value. Constraints (18) represent the binary constrains.

5. Numerical Experiments

discussions are shown in Section 5.2.

5.1 Design of Experiments

The experimental design is separated into two phases. In phase one, we generate the related parameters of the mathematical model. In phase two, we create the problem instances.

5.1.1 Parameter generation and description

Firstly, we set the cost coefficients 𝐶_I, 𝐶_J in this paper. Traveling costs 𝐶_I can be viewed as the costs of electricity usage. The price of electricity usage is €0.08/𝑘𝑊ℎ (Centraal Bureau voor de Statistiek, 2015). Moreover, we set the fixed costs 𝐶_J of using one ECV as €53.97 per day, which is assumed based on the work of Van Duin, et al., (2013).

Secondly, the parameters and assumptions related to the EVC for transportation are based on the prior research of Iwan, et al. (2014). We opt for Streetscooter Work to test the proposed model in this study, due to its low battery recharging time and low acquisition price. The characteristics of Streetscooter Work is shown in Table 5.1. Further, the recharging time rate 𝑔 is calculated as 300/ 80 = 3.75 mins/ km. According to the work of Schneider, et al. (2014), the battery consumption rate 𝑐 is set as 1 in order to reduce the complexity of the calculations in this paper.

Name Streetscooter Work

Manufacture Spijkstaal Elektro B.V.

Carrying Capacity (kg) 700

Carrying Capacity (m3) 4.3

Maximum velocity (km/ hour) 85

Travel Range (km) 80

Battery Charging Time (minutes) 300

Price (EUR) 31,775

Table 5.1 Characteristics of Streetscooter Work (Iwan, et al., 2014)

dummy nodes should be set as small as possible to reduce the grid size, but large enough to avoid the restriction of potential multiple visits. We assume that the number of dummy nodes is equal to the number of customers so that there is one potential visit to charging station associated with one customer. Therefore, each vehicle has the possibility to recharge after servicing each customer.

Fourthly, 50 is the maximum operational speed (kilometer/ hour) of a vehicle when traveling inside the city Groningen (Buck Consultants International, 2013). However, the street in Groningen is narrow, so that we assume 30 kilometers/ hour is the average speed of the ECV to travel inside Groningen city. Further, we assume 60 is the average operational speed (kilometer/ hour) of the ECV when traveling between the depot and city area. This assumption is made after considering other influential factors, such as the maximum operations speed of the ECV, congestion and weather conditions etc. We also set 9 minutes is the average service time s_’ at any customer location i (Bhusiri, et al., 2014). Therefore, the traveling time t_’” between different nodes is categorized as in Table 5.2. We use Euclidean distances to calculate the travel distance 𝑑_CF between any two nodes.

From/ To Depot Customers Charging station

Depot -- 𝑑𝑖𝑗 60×60 𝑑_𝑖𝑗 60×60 Customers 𝑑𝑖𝑗 60 ×60 𝑑_𝑖𝑗 30×60 𝑑_𝑖𝑗 30×60 Charging station 𝑑𝑖𝑗 60 ×60 𝑑_𝑖𝑗 30×60 --

Table 5.2 Calculations of traveling time

Finally, we assume the time window of customers is sparsely distributed and it should allow the possibilities of generating flexible time window. Thus, we assume the earliest allowable time l_’ for service at customer node i is randomly distributed between 05:00 and 11:00. The latest allowable time u_’ = l_’+ 60 minutes, considering the delivery agreement with customers in reality. In addition, referring to the work of Taş, et al. (2014), fractions 𝑝_CO _{and 𝑝}

CP are set to allow the maximum violations, and we employ 1 as a symmetric violation in this paper, which means 𝑝_CO _{= 𝑝}

In this study, all of the parameters generated in phase one are kept as fixed numbers in all the problem instances created in phase two. The value of each parameter is summarized in Table 5.3. Parameter 𝐶_I 𝐶_J 𝐶 𝑄 𝑔 𝑐 𝑠_C 𝑝_C Value €0.08/ kWh €400 4.3m 3 _80km 3.75 mins/ km 1 9mins 1

Table 5.3 Value of parameters

5.1.2 Overview of problem instances

In phase two, we create problem instances to assess the quality of solutions. We consider all the instances in one-day operations with one ECV. Initially, four instances are created based on the different number of customer and amount of total customer demands. In order to generate different instances in the experiment, the number of customers are set as 6, 7, 8, 9 respectively.

Methods of generating location of nodes. According to Buck Consultants International

(2013), there are 120,000 square meters (120 km2) allocated to shops in city Groningen. Hence, we generate the locations in all scenarios as following: 1) There is one depot locates in the south of the city, and it is 15 kilometers away from the city center. 2) The datasets of customer locations are designed to be randomly generated. X-axis and y-axis of the customer nodes are generated within a grid of 11 by 11 kilometers, which is similar in size of city Groningen. 3) There is one charging station in the problem instances, which is located in the center of the grid.

Description of scenarios under each instance. Referring to the problem description in

Figure 3.2, there are four types of configuration. Hence, they are applied in each instance as four different scenarios that we mainly address in this experiment. The general characteristics of these four scenarios are described as following.

• RnF: there is one charging station in the city center to allow the recharging possibilities, and vehicles are allowed to recharge in the depot as well. At the same time, vehicles should serve the customers within their original time window.

• nRF: contrary to RnF, vehicles are allowed to serve the customer within the flexible time window, but vehicles are not allowed to recharge in the charging station in the city center.

• RF: vehicles are allowed to recharge in the charging station located in the city

center, and serve the customers within their flexible time window.

In conclusion, we create four instance types regarding the numbers of customer nodes associated with different location distribution. Each instance has between six and nine customer nodes, which represent arbitrarily selected delivery locations in Groningen city. Hence, the instances with the same number of customer nodes are tested in the identical geographical dispersion. Under each instance, four scenarios are generated based on different combination of recharging and flexible time window possibilities. Therefore, we have 16 instances in total to test the model. Table 5.3 summarizes all the problem instances apply in this paper.

Instance #Customers Total demand (m3)

Charging station and flexible time window possibilities

1 6 4 nRnF, RnF, nRF, RF

2 7 4.2 nRnF, RnF, nRF, RF

3 8 4.1 nRnF, RnF, nRF, RF

4 9 3.9 nRnF, RnF, nRF, RF

Table 5.3 Problem instances description

(Code description: referring to Figure 3.2, nRnF denotes to type I, RnF denotes to type II,

nRF denotes to type III, and RF denotes to type IV.) 5.2 Results and Discussions

5.2.1 Computational time limit

exceptions of scenario RnF in instance 3 and 4. Although Mosel can provide the best bound of both instances with a gap of 100%, we cannot obtain the optimal solution within 7200 seconds. Additionally, we stop running the model in 7200 seconds, due to the time-consuming problem. Notice that, there is no feasible integer solution exists in most of the scenarios, and we will discuss the results in the Section 5.2.2. In below, Table 5.4 shows the computational time of each instance.

nRnF RnF nRF RF

Inst. f Obj. t(s) f Obj. t(s) f Obj. t(s) f Obj. t(s)

1 - - <0.1 - - 2.4 - - 0.1 95 62.445 0.4 2 - - 0.1 - - 96.3 - - <0.1 96 64.775 1.1 3 - - 0.1 - - - 1.2 96 62.65 11.3 4 - - 0.2 - - - 4.6 89 64.59 53.1 Table 5.4 Computational time for problem instances

(Notes: f denotes the total travel distance, Obj. the objective function, and t(s) the total running time in seconds.)

5.2.2 The effect of allowing recharging in the city center and flexible time window

Results of scenarios nRnF, RnF and nRF. As we can see in Table 5.4, applying the first

three scenarios (nRnF, RnF, nRF) in all the four instances results in no feasible integer solution. In other words, the model cannot find any solution to complete the delivery with given parameters and scenarios setting. Arbitrary parameter can be a factor leading to this result, including the number of ECV, travel distance, duration of recharging in the case of RnF, and the size of time window in the case of nRF. Therefore, it is necessary to modify one or more than one of these parameters in order to complete the delivery in these three scenarios, such as modifying the number of ECV to two. Later in this section, there is an analysis of results in the case of using two ECV in the experiment.

Results of scenarios RF. This is the only scenario that the model is successful to output a

Applying two ECV. As discussed above, using only one vehicle cannot solve the routes

planning issues in scenario nRnF, RnF, and nRF. Thus, we alter the vehicle number from one to two to have further investigation. Table 5.5 list the results of each instance.

nRnF RnF nRF RF

Inst. f Obj. f Obj. f Obj. f Obj.

1 m=1 - - - 95 62.445 m=2 94 115.46 94 115.46 94 115.46 94 115.46 2 m=1 - - - 96 64.775 m=2 99 115.86 99 115.86 99 115.86 99 115.86 3 m=1 - - - 96 62.65 m=2 96 115.62 96 115.62 96 115.62 96 115.62 4 m=1 - - - 89 64.59 m=2 107 116.5 107 116.5 107 116.5 107 116.5

Table 5.5 Results of total travel distance and total transportation costs

(Notes: m denotes the number of ECV used, f denotes the total travel distance, and Obj. the objective function.)

As shown in Table 5.5, in all four instances, using two ECV is able to trigger the model to finish the transportation in the first three scenarios (nRnF, RnF, nRF) in which the model fails to provide any feasible solution if the number of EVC is one. The results are interpretable. Due to the failure of initial battery capacity covering all the routes, the ECV needs to be recharged during the servicing. However, both traveling and recharging operations consumes time, and vehicle lost the flexibility to serve the customers arriving either earlier or later. Hence, one more vehicle is required to satisfy all the customers within their time windows. Whereas, increasing the number of vehicle used inevitably incur the total transportation costs increase 1.8 times compared to the one vehicle case. At the same time, the utilization rate decreases from 100% to 50%.

Figure 5.1 Total costs and travel distance of instance four

A notable fact derived from Table 5.6. The recharging operations, the total earliness of servicing and the total delay of servicing are all zero in the case of using two ECV under all instances. Same to the results in Table 5.5, in the same instance, using two ECV in this experiment outputs no different solution no matter in which scenario. The results suggest that it is unnecessary to have recharging possibilities in the city center and customer’s flexible time window allowance when using two vehicles. This is because the battery capacity of two vehicles is sufficient to cover all the battery requirement for the routes. Consequently, this leads to no need to serve the customer earlier or later in order to have recharging operations.

Table 5.6 Results of number of recharging operations, earliness, and delay

(Notes: r denotes to number of recharging operations, e the total earliness arrival time at customers in minutes, l the total delay arrival time at customers in minutes.)

€ 64.59

€ 116.5 89

107

m=1 m=2

Total Costs and Travel Distance of Instance Four

All in all, lack of either possibility of recharging in the city center or allowance of customer flexible time window will lead to no solution in this experiment. Both factors are significantly influencing the transportation costs. The results suggest that it is necessary to both allowing recharging in the city center and flexible time window of customer. By allowing both factors, a reduction in the number of vehicles used, a decrease in the travel distance, and improve vehicle utilization rate can be achieved.

6. Conclusions

In this paper, we explore the electric vehicle routing problem in urban freight transportation. We introduce the opportunities in vehicle route plans with stops inside the city center for recharging operations in case of eliminating the risk of running out of battery. Simultaneously, we present the possibilities of customer’s flexible time window to enable the earliness and delay servicing to relax the restriction of customer’s time window.

The contributions of this paper are twofold: firstly, it tries to enrich the existing literature by optimizing the routes for electric vehicles with recharging possibilities inside the city center, involving the concern of customer’s flexible time window. Secondly, in practice, the developed mathematic model aims to support the transportation companies to conduct eclectic vehicle routing planning when customers allow implementing flexible time window to receive goods.

and customer’s flexible time window. Corresponding to Figure 3.2, compare with the case of nRnF, RnF, and nRF, a more flexible routes planning can be achieved by RF. Both allowing recharging in the city center during service and implementing customer’s flexible time window, provide more options for decision making in conducting recharging operations. This leads to greater service range of single vehicle as well. The results and conclusions of this experiment may also apply in the cities that have the similar size as the city of Groningen.

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