"A multi-trip truck scheduling model for inland container terminals with hard time-windows, service times and European driving regulations"

"A multi-trip truck scheduling model for inland container terminals with hard time-windows, service times and European driving regulations"

June 30, 2020

Author: Jasper Onderwater S3526534 Supervisor: dr. ir. S. Fazi Second Supervisor: dr. C. Xiao

Master Thesis MSC TOM Technology Operations Management

Faculty of Economics and Business University of Groningen, Netherlands

Word-count: 7886

The biggest challenge for inland container terminals is the pre- and end- haulage of containers . To improve efficiency of these drayage operations a model is designed to combine the delivery and pick-up of empty and full containers at different customer locations during a trip. The model takes time-windows at customer locations, European driving regulations, multiple trips and the service time into consideration and aims to minimize the total driving time. The model is tested with the use of CPLEX and is able to handle small instances fast. After a sensitivity analysis the model is adjusted to also minimize the number of drivers to increase the average driving time per driver.

Keywords: VRP, Vehicle Routing Problem, European Driving Regu-

lations, Delivery and Pickup, Hard time-windows, Multi-trip, Inland Con-

tainer Terminal, Drayage

First of all I would like to thank my supervisor dr. ir. S. Fazi who helped

me during the development of my thesis. I also would like to thank my

father, family and my friends who supported me during the writing of my

thesis, especially during these times of COVID-19.

1 Introduction 1

2 Literature Review 2

2.1 VRP . . . . 3

2.2 FTVRP . . . . 3

2.3 PDP . . . . 4

2.4 FTPDPTW . . . . 5

2.5 MTVRP . . . . 6

2.6 VRPDWH . . . . 7

2.7 European Driving Regulation . . . . 9

3 Problem Description 10 4 Mathematical model 12 4.1 Formulation . . . . 13

5 Numerical Experiments & Results 20 5.1 Data generation . . . . 20

5.2 Input and output of the model . . . . 22

5.3 Experiments . . . . 23

5.4 Setting . . . . 26

5.5 Results . . . . 26

6 Discussion & Limitations 28 6.1 Discussion . . . . 29

6.2 Limitations . . . . 31

7 Conclusion & Future Research 32 7.1 Conclusion . . . . 32

7.2 Future research . . . . 33

1 Introduction

In today’s economy container transport is a common way of transporting goods around the world for B2B and B2C. Container transport is steadily growing be- cause of economic globalization Frémont and Franc (2010) therefore transport of goods in containers is expected to grow from 23.23 MTEU in the start of 2020 to 24.05 MTEU at the end of 2020 this is an increment of 3.5% (Unctad, 2019;

Wagner, 2020).

One of the biggest challenges in container transport is the pre- and end- haulage (Caris & Janssens, 2009). During the end-haulage containers needs to be trans- ported from the port to the customer located in the hinterland and vice versa dur- ing the pre-haulage. Containers are moved by barge, trains or trucks from or towards strategically placed inland container terminals. In many cases the last mile-haulage is performed by trucks since many customers do not have direct access to a waterway or a railroad (Van de Lande et al., 2018). Containers arrive from overseas at the sea terminal and are unloaded from container vessels. The ar- rived containers are waiting to be loaded on barge, train or truck to be transported towards the inland terminal. Once containers arrive in the hinterland, containers are need to be moved between the inland terminal and the customer or vice versa as depicted in 1.1.

Figure 1.1: Movement of containers The focus of this research is the

transport of containers between the in- land terminal and customers by truck.

Basically, trucks only have two tasks, to deliver and pickup containers at the terminal and customers. Initially, trucks start at the inland terminal and deliver full or empty containers to the customer and returns towards the in-

land terminal. To increase efficiency trucks can visit multiple customers during a trip which makes the schedule more complicated.

In the process of scheduling the containers are allocated to drivers with trucks

and trips. Due to the presence of: (1) Time-Windows for delivery , (2) European

driving regulations and (3) full and empty containers at customer locations it is a serious challenge to come up with a planning which minimizes the overall costs.

The literature lacks a model which considers the transport of full and empty con- tainer between customers and the terminal, delivery in specified time-windows, multiple-trips and comply to European driving regulations. Important reasons to incorporate regulations into scheduling is to: (1) increase road safety, (2) increase accuracy of the planning and (3) comply to European legislation. Accidents can lead to serious injuries, fatalities, environmental damage and economic damage.

Several papers in recent years are concerned about vehicle routing and driver reg- ulations (Archetti & Savelsbergh, 2009; Goel, 2010; Goel & Irnich, 2017). This paper will be concerned about scheduling the transport of empty and/or full con- tainers from the terminal to the customer or between customers. The aim of the model is to increase efficiency of the trips by allowing the truck to visit multiple customers during one trip and using a truck and driver in multiple trips. In our problem there is a distinction made between customers requiring an empty or a full container, more details about the problem can be found in the problem de- scription. The aim of the paper is to come up with a mathematical model, the purpose of such a model is to improve scheduling decisions and reduce costs by minimizing the total time driven to visit each customer. In practice, this planning model can help a planner to decrease his time to form scheduling decisions, de- crease the costs for hiring/firing and the salary and thus improve the profit margin of the company.

The structure of this thesis is as follows: section 2 will review the literature related to vehicle routing and European safety regulations for truck drivers. In sec- tion 3 the problem is described and in section 4 mathematical model is presented.

Section 5 will show the results of the mathematical model, while section 6 the results and limitations are discussed. The last section will end with a conclusion and recommendations for future research.

2 Literature Review

In the VRP literature there are many variants available. The first section will

discuss the VRP variants which are closely related to the proposed problem. In

the second section an overview will be given about the VRP applicable for full truckload scheduling. The last section will clarify the European driving regula- tions.

2.1 VRP

The earliest study considering a capacitated vehicle routing problem (C-VRP) is studied by Dantzig and Ramser (1959) considers a problem, in which trucks had to deliver gasoline from a bulk-terminal towards multiple gas stations. The goal is to minimize the total distance and satisfy the requested demand of the gas stations; trucks got a constraint for the quantity gasoline carried. The goal of the VRP is to find a trip against the least-cost in a way each customer is visited once, vehicles start from a depot and end in the depot and the capacity of vehicles is not exceeded. This form is the simplest and most studied VRP and is described as follows: “The problem of designing least cost delivery trip from a terminal to a set of geographically dispersed customer locations subject to a set of constraints”

(Kumar & Panneerselvam, 2012). Many additions on the original VRP exist, in the next part the additions are discussed which are interesting for the proposed problem.

2.2 FTVRP

Full truckload routing problems (FTVRP) are closely related to C-VRP prob- lems in a C-VRP the load can be split and the truckload can consist of more than 1 unit and is therefore considered as less than truckload (LTL). In a FT the load cannot be split and the truck can only carry one unit of cargo between the nodes.

A FT-VRP is typically used for scheduling trucks serving container terminals. In

container terminals there are outbound and inbound logistics of containers to the

terminal or customer, these containers are full or empty so the capacity of the

truck can never be more than one (El Bouyahyiouy & Bellabdaoui, 2017; Nos-

sack & Pesch, 2013). The first routing problem involving FT is introduced by

Ball, Golden, Assad, and Bodin (1983) and solved by applying a heuristic algo-

rithm. Their purpose of the study was to provide managerial insights about the

amount of trucks which should bought or leased. After the introduction of FT in

routing problems many variants to this problem exist, for our research we focus on FTVRP which are considering: pickup and delivery, time windows, driving regulations and multi-trips.

2.3 PDP

A pickup and delivery problem (PDP) is a problem in which people and/or goods are transported between the origin and destination and can be picked up and delivered at different nodes. A widely accepted concept of PDP comes from the researchers Berbeglia, Cordeau, Gribkovskaia, and Laporte (2007) and argues a PDP can come in three variants: one-to-one (1-1), one-to-many-to-one (1-M-1) and one-to-many-to-many (1-M-M).

The 1-1 variant can be seen in figure 2.1 the truck start at the terminal and picks up goods at the customer, and deliver the goods to the next customer. This problem can be found in courier operations where only deliveries takes place, no goods are transported towards the terminal.

Figure 2.1: PDP Variant 1-1

The 1-M-1 variant considers delivery and pick-up nodes, goods are initially

delivered from the terminal an example can be seen in figure 2.3. The numbers

above the arrows resembles the amount of goods in the transport unit. The neg-

ative number resembles the amount to be delivered and the positive number re-

sembles the amount ready for pickup. The initial transport of goods starts at the

terminal towards the customer, if the customer has any goods available for pickup

it will be transported to the terminal. This problem can be found in container

transport where full containers are delivered to the customer and empty or full

containers are loaded at the customer and transported back to the terminal. In

literature this problem is addressed as a VRP with back-hauls, where first all de- liveries are performed and thereafter all goods are picked up at customer locations before the truck returns to the terminal. Wang, Mu, Zhao, and Sutherland (2015) considers a VRPSPDTW, in this problem trucks can perform a delivery and a pickup simultaneously at a customer node. When all goods carried from the depot are delivered the gathered picked up goods are transported to the depot. In our problem this can be done with full containers, however we also consider empty containers in our problem which can be exchanged between customers.

Figure 2.2: PDP Variant 1-M-1

The 1-M-1 variant implies that a node can resemble a pick-up and a delivery point. Figure 2.3 shows how this variant may work, it starts from the terminal with or without any goods it will perform a delivery and a pickup at the first customer and deliver the picked up goods to the second customer, when the truck is empty the truck returns to the terminal.

Figure 2.3: 1-M-M

In our problem we consider a mix, for the delivery and pickup of full con- tainers the 1-M-1 variant and for the delivery and pickup of empty containers the 1-M-M variant.

2.4 FTPDPTW

A popular addition to a VRP is a time-window in which goods are expected

to be delivered to each customer Solomon (1987). Time-windows in VRP’s are

important to comply with a real world problem (Baldacci, Mingozzi, & Roberti, 2012). The VRPTW (Vehicle Routing Problem with Time Window) does not only considers the shortest trip and the capacity of the truck but also a time window in which goods are expected to be delivered. Arunapuram, Mathur, and Solow (2003) proposed a solution for a full truckload problem with multiple depots and time windows (M-FTPDPTW). In the research the problem was solved by in- troducing a branch-and-bound system that embeds column generation which is solved by integer programming. Zhang, Yun, and Kopfer (2010) introduced pick- up and delivery problem with multiple depots and time windows (M-FTPDPTW) and formulated as a multi-traveling salesman problem. The problem is solved by applying a reactive tabu search heuristic. The authors Nossack and Pesch (2013) improved the method used by Zhang et al. (2010), and used a 2 stage heuristic algorithm to solve the problem. Sterzik and Kopfer (2013) introduced a M-FTPDPTW which integrates the allocation of empty containers between ter- minals and solved this by applying a tabu search Heuristic. The paper of Caris and Janssens (2009) also considers FTPDPTW and solved the problem with a two phase heuristic and considers merging trips such that a backhaul can be performed.

Caris and Janssens (2009) only consider one terminal, while other FTPDPTW problems considers multiple terminals (Arunapuram et al., 2003; El Bouyahyiouy

& Bellabdaoui, 2017; Nossack & Pesch, 2013) or separate depots dedicated for empty containers (Zhang et al., 2010). All these researches do not consider the addition of Multiple Trips, driving regulations and will return the depot/terminal after an pickup and always serve the customers from the depot or terminal.

2.5 MTVRP

An extra addition to VRP problems are the VRP’s with multi-trips (MTVRP),

the addition of a MT formulation to a VRP is the use of vehicles multiple times

(Taillard, Laporte, & Gendreau, 1996). For example in practice MT is applicable

if there is a limited number of vehicles available. Researchers Azi, Gendreau, and

Potvin (2010) proposed the first exact algorithm in which vehicles can be used

multiple times (MTVRPTW), they use a column generation approach in which

not all customers can be visited due to a limited number of available vehicles.

Hernandez, Feillet, Giroudeau, and Naud (2014) formulated a MIP (mixed integer program) for a MTVRPTW the formulation contained service times and has soft time windows which means a truck can arrive before the opening of the time- window. The research of Hernandez et al. (2014) is closely related to our problem but considers only a 1-M-1 and lacks the driving regulations.

2.6 VRPDWH

There are researches available which do incorporate driving regulations and time

windows (Goel, 2010; Goel & Irnich, 2017) which is presented as a VRP with

truck driver scheduling (VRPTDS). Archetti and Savelsbergh (2009) added drivers

working hours to the FT problem with Time windows (FTDWH). These researchers

do not consider Multi-trips and the combination of 1-M-1 and 1-M-M. Table 2.1

shows an overview of the differences between the papers.

Author

&Date Problem

>1 Terminal/depot

Combined 1-M-1

&1-M-M Simultaneous Pickup/Deli

very

windo Time

ws

ving Dri

regulations Multi-trip

Ball, Golden, Assad & Bodin (1983) FTVRP No No No No No No Arunapuram, Mathur , & Daniel (2003) M-FTPDPTW Y es No Y es Y es No No Caris & Janssens (2009) FTPPDPTW No No No Y es No No Archetti & Sa v elsber gh (2009) VRPD WH Y es No No Y es Y es No Zhang, Y un & K opfer (2010) M-FTPDPTW Y es No Y es Y es No No Azi, Gendreau,and Potvin (2010) MTVRPTW No No No Y es No Y es Nossack & Pesch (2013) M-FTPDPTW Y es No Y es Y es No No Sterzik and K opfer (2013) M-FTPDPTW Y es No Y es Y es No No Hernandez et al. (2014) MTVRPTW No No No Y es No Y es W ang et al. (2015) VRPSPDTW No No Y es Y es No No Goel, & Irnich (2017) VRPTDS No No No No Y es No El Bouyah yiouy & Bellabdaoui (2017) FTPDPTW Y es No Y es Y es No No This research MTFTPDPTW No Y es Y es Y es Y es Y es T able 2.1: Ov ervie w of literature

All studies found are different from the proposed problem since these studies do not combine the following points: The pickup of an empty container and de- liver this empty container to the nearest customer which is demanding an empty container, the delivery and pick up of full containers and hard-time windows, multi-trips and the European driving regulations.

2.7 European Driving Regulation

One of the first constraints in the model are the driving times, these driving times are relevant for all truck driver which drive a vehicle which exceed a total load of 7.5 tons European Union (2006).

REGULATION (EC) No 561/2006 for driving times

The regulation for driving times affect all drivers who perform road haulage operations, in this regulation the maximum driving and minimum rest times are anchored. The maximum hours of driving times are separated into: contiguous-, daily-, weekly- and biweekly driving times. In addition, the driver has restrictions how long the driver can contiguously drive, and has to take a rest or a break.

Breaks

During a break, the driver cannot perform any tasks such as unloading, load- ing, technical maintenance, etc. this is counted as work time. Roughly, the activi- ties of a driver can be split-up into driving and other tasks. One of the constraints are the minimum resting times and the frequency of these. After a contiguous drive of 4.5 hours, a driver must have a break of 45 minutes. A driver can split his 45 minutes break into 15 minutes and 30 minutes. E.g. a 2 hours’ drive, 15 minutes break followed by a 2.5 hours’ and a 30 minutes break. A driver must have at least 11 hours of contiguous rest per day. In addition, Per week a driver must have at least 45 hours of contiguous rest, and can be reduced to 24 hours every second week. The definition of a week is from Monday 0 pm - Sunday 0 pm. The daily rest time must be at least 11 hours; however, this can be split up into 3 hours of contiguous rest, followed by 9 hours of contiguous rest.

Driving times

A worker is only permitted to drive 9 hours per day; however, a driver can

stretch his driving time 2 times a week to 10 hours. Per week, a driver cannot exceed a total of 56 hours and in two consecutive weeks; the driver must not exceed 90 hours.In addition, there are restrictions on the maximal amount of hour a worker can work during a night shift. In addition, when the driver works between 22 pm and 6 am the break of 45 minutes must be kept before a 3-hour contiguous drive. A night shift is considered if a driver works between 12 pm and 6 am.

During a night shift the driver is allowed to work 43 nights in 16 weeks or 20 hours per two weeks. Per night, the law permits a driver to work a maximum of 10 hours in a period of 24 hours.

3 Problem Description

The goal is to tackle the problem of planning a set of customers to feasible trips and available drivers and trucks. The trucks and their drivers always originate from a single terminal. The trucks and their drivers can only serve the customer locations during specified hard time-windows, the demand for these customers are known one-day in advance. There are two types of customers specified which can have the following types of requests:

Importer

1. Request a delivery of a full container and no release of container.

2. Request a delivery of a full container and release a full container.

3. Request a delivery of a full container and releases an empty container.

Exporter

1. Request a delivery of an empty container, no release of a container.

2. Request a delivery of an empty container, release of a full container.

3. No delivery request of a container, release a full container.

Customers who request a full container are always served from the terminal.

Customers which are requesting an empty container can be served by another

customer who releases an empty container or the terminal. Trucks which serve a

customers which release a full container must drive back to the terminal to drop

off the full container before performing a new. The trucks can only carry one container per run and are coupled to drivers because the time-windows are only opened during the day a driver cannot reach more than 9 hours in our problem.

An example of trips which can be performed can be found in figure 3.1.

Figure 3.1: An example of the possible trips.

In the problem three types of trips are considered:

1: A truck loads a full container at the terminal and transports this container to a delivery node and performs a pickup of a full container and returns to the terminal.

2: A truck loads a full container at the terminal and transports this container to a delivery node and makes a pickup of an empty container at this node and drives to another customer or terminal.

3. A truck transports a full container to the delivery node and drives empty to the next pickup point where a full container is loaded and transported to the terminal.

4. A customer only needs a pickup, trucks which only performed a delivery of a container can service this node.

In case 3 and 4 trips should be merged as this done in the paper of Caris and

Janssens (2009), to reduce the time a truck travels empty see figure 3.2.

Figure 3.2: Merge trips as in Caris and Janssens (2009)

Driving hours are considered in the problem, the authors Archetti and Savels- bergh (2009) even consider the driving regulations as a crucial complexity. In our problem the driver is allowed to drive for a maximum of 9 hours per day and must take a rest of at least 45 minutes before he reached a continuous drive of 4.5 hours.

The objective of the formulated model is to find the optimal assignment of trips to drivers this is done by minimizing the total distance of serving all customers without violating the European driving regulations and time-windows.

4 Mathematical model

The mathematical model aims to deal with the problem of assigning customers

to trips and the trips to drivers with trucks. Also the demand of customers must

be satisfied and the driving regulations must be obeyed, the model considers a

round trip which start at the terminal and ends at the terminal. The trips are added

to use vehicle multiple times during the shift, the model will search for the op-

timal solution by generating a solution with the minimum amount of travel time

to serve all customers, while the containers must be delivered between specified

time-windows.

4.1 Formulation

The model is based on a directed graph consisting of G = {V, A}, V describes the set of nodes and A = {(ij)|ij ∈ V : i 6= j} describes the set of arcs between the nodes. Besides the set of customers, there are time-windows considered in which the customers expect the visit of a truck between the earliest service time a _i and the latest service time b _i . The time windows are hard constraints this means if the vehicle arrives before the earliest service time the truck must wait. On the other hand if the vehicle arrives to late the trip is infeasible. Customers also have a service time s i this is the time spent by the vehicle to load and unload the containers.

Customers have specific demands, when a customer request the delivery of a full container q ^df _i the container can only originate from the terminal {0}, whereas a customer request the pickup of a full container q _i ^pf the driver will return to the terminal {n + 1}. Customers who require the delivery of an empty container q _i ^de can either be served from the terminal {0} or after a pickup of an empty container q _i ^pe at a customer. In the case there is no pickup at customer i, the driver can visit customer j which only requires a pickup of a container. The capacity of the is one container when traversing over arc ij, if the truck hauls an empty container h ^r _ij equals one. In the case the truck hauls a full container ˆ h ^r _ij will equals one, when there are no containers both binary indicators will be 0.

The drivers must obey the European driving regulations, which means a driver must rest for 45 minutes (β) before reaching a continuous drive of 4.5 hours (α) and cannot drive more than 9 hours per day. The continuous drive time consist of the accumulative time of driven trips, this includes travel time τ _ij and the service time s _i . The travel times should satisfy the triangular inequality: τ _ij + τ _jk ≥ τ _ik .

Because vehicles are used multiple times during a shift, the set of trips R is added. During a trip at least one customer is visited, so the number of trips must be less or equal to the number of customers |R ≤ N |.

Sets

N : {1 . . . , n} is the set of customers ranging from 1 to n.

V : N ∪ {0, n + 1}is the set of nodes, in which N represents the customers and

{0, n + 1} the terminal.

A: is the set of arcs, composed by a tuple of of nodes which exist in the direct graph {G = V, A}.

R: is the set of trips.

D: Represent the set of available drivers.

Parameters

τ ij : the matrix in which the time is associated when visiting customer i to j q _i ^pf : is the amount of full containers that will be picked up at node i

q _i ^pe : is the amount of empty containers that will be picked up at node i q _i ^df : is the amount of full containers that will be delivered at node i q _i ^de : is the amount of full containers that will be delivered at node i a _i : Openings time of the time window for customer i

b _i : Closing time of the time window for customer i s _i : Service time at each customer, including the terminal β: Resting time for drivers in hours

α: Maximum amount of continuous driving hours

Decision variables

x ^r _ij : Binary indication 1 if truck with a driver is traversing from i to j, 0 other- wise.

h ˆ ^r _ij : Binary indication 1 if a truck is assigned to trip r traveling trough arc: {ij}

with a full container, 0 otherwise.

h ^r _ij : Binary indication 1 if a truck is assigned to trip r traveling trough arc: {ij}

with an empty container, 0 otherwise.

σ _i ^r : Binary indication 1 if customer i is visited during trip r, 0 otherwise p ^r _d : Binary indication 1 if driver d is assigned to trip r, 0 otherwise.

z _rr ^d

: Binary indication 1 if driver d carries out trip r ⁰ immediately after trip r v ^di _r : Binary indication will turn 1 if driver d rests during trip r after serving node i, 0 otherwise.

S _ri ^start : Continuous time variable to track the starting time of the service in trip

r at a customer.

S _ri ^end : Continuous time variable to track the ending time of the service in trip r at a customer.

dt ^dr _r : Continuous time variable to track the accumulative driving time for each driver in each trip.

Objective function

M inimize X

i,j∈A

X

r∈R

x ^r _ij τ _ij (1)

S. t.

X

j∈V :i6=j

X

r∈R

x ^r _ij ≤ 1, ∀i ∈ V \ {0} (2)

X

j∈V :i6=j

x ^r _ij − X

jV :i6=j

x ^r _ji = 0, ∀i ∈ V \ {0, n + 1}, r ∈ R (3)

X

j∈V \{n+1}

x ^r _0j = max

ij∈A x ^r _ij , ∀r ∈ R (4)

X

i∈V \{0}

x ^r _i,n+1 = max

ij∈A x ^r _ij , ∀r ∈ R (5)

X

r∈R

x ^r _ij + X

r∈R

x ^r _n+1,0 + X

r∈R

x ^r _0,n+1 = 0, ∀ij ∈ V \ {i = j} (6)

h ^r _ij + ˆ h ^r _ij ≤ x ^r _ij , ∀ij ∈ A, r ∈ R (7)

X

i∈V

X

r∈R

h ^r _ij − q ^de _j + q _j ^pe = X

iV

X

r∈R

h ^r _ji , ∀j ∈ V \ {0} (8)

X

i∈V

X

r∈R

ˆ h ^r _ij − q ^df _j + q _j ^pf = X

iV

X

r∈R

ˆ h ^r _ji , ∀j ∈ V \ {0} (9)

X

r∈R

x ^r _0i ≥ q _i ^df , ∀i ∈ V \ {0} (10)

X

r∈R

x ^r _i,n+1 ≥ q _i ^pf , ∀i ∈ V \ {0} (11)

X

j∈V :i6=j

x ^r _ij = σ _i ^r , ∀i ∈ V \ {0}, r ∈ R (12)

X

d∈D

p ^r _d = max

i∈V \{0} σ _i ^r , ∀r ∈ R (13)

p ^r _d + p ^r0 _d − z _d ^rr0 ≤ 1, ∀, d ∈ D, r ∈ R, r0 ∈ R : r0 < r (14)

S _ri ^start + s _i + X

d∈D

(βv ^di _r ) + τ _ij − S _rj ^start ≤ (1 − x ^r _ij )M, (15)

∀ij ∈ A, d ∈ D, r ∈ R

S _ri ^start + s _i + τ _ij − S _rj ^start ≥ (x ^r _ij − 1)M, ∀ij ∈ A \ {i = 0}, r ∈ R (16)

S _ri ^start + s _i + τ _ij − S _rj ^start ≥ (x ^r _ij − 1)M, ∀ij ∈ A \ {j = n + 1}, r ∈ R (17)

a _i ≤ S _ri ^start ≤ b _i , ∀i ∈ V \ {0}, r ∈ R (18)

S _ri ^end = S _ri ^start + s _i , ∀i ∈ V, r ∈ R (19)

S _r,n+1 ^end + βv ^d _n+1,r − S _r0,0 ^start ≤ ((1 − p ^r _d ) + (1 − p ^r0 _d ) + (1 − z _d ^rr0 ))M, (20)

∀d ∈ D, r ∧ r0 ∈ R : r < r0

dt ^dr0 ₀ ≤ X

r∈R:r<r0

z _d ^rr0 M, ∀d ∈ D, r ⁰ ∈ R (21)

dt ^dr _i ≤ S _ri ^end , ∀d ∈ D, r ∈ R, i ∈ V (22)

max i∈V (dt ^dr _i ) ≤ p ^r _d M, ∀ d ∈ D, r ∈ R (23)

dt ^dr _i − αv ^dr _i + τ _ij + s _j − dt ^dr _j ≤ ((1 − x ^r _ij ) + (1 − p ^r _d ))M, (24)

∀d ∈ D, r ∈ R, ij ∈ A

τ _ij + s _j − dt ^dr _j ≤ ((1 − x ^r _ij ) + (1 − p ^r _d ))M, ∀d ∈ D, r ∈ R, ij ∈ A (25)

dt ^dr _n+1 + s ₀ − dt ^dr0 ₀ − αv _n+1 ^dr ≤ ((1 − z _d ^rr0 ) + (1 − p ^r _d ) + (1 − p ^r _d

))M, (26)

∀d ∈ D, r ∧ r0 ∈ R

X

r∈R

X

i∈V

v ^dr _i ≤ 1, ∀d ∈ D (27)

P i∈V P r∈R max dt ^dr _i ≤ α, ∀d ∈ D (28)

dt ^dr _i ≥ 0, ∀d ∈ D, r ∈ R, i ∈ V (29)

S _ri ^start , S _ri ^end ≥ 0, ∀r ∈ R, i ∈ V (30)

x ^r _ij , ˆ h ^r _ij , h ^r _ij ∈ {0, 1}, ∀ij ∈ A, k ∈ K, r ∈ R (31)

p ^r _d , z _d ^rr

∈ {0, 1}, ∀r ∧ r ⁰ ∈ R, d ∈ D (32)

σ _i ^r ∈ {0, 1}, ∀r ∈ R, i ∈ V (33)

v ^dr _i ∈ {0, 1}, ∀d ∈ D, r ∈ R, i ∈ V (34)

The objective function (1) aims to minimize the total travel distance of serving all customers which require a pickup or/and a delivery of an empty or full con- tainer. Constrain (2) ensures a customer is only visited once, (3) ensures that each truck with a driver will leave the node after visiting. Constraint (4) guarantee each trip will start at the terminal and (5) enforce each trip which leaves the terminal will return to the origin {n + 1}. (6) Is necessary to ensure the truck and a driver will not travel between the same node.

The following constraints ensure the pickup and delivery of containers, restric-

tion (7) ensures a truck will never carry more than one container when traversing

arc ij. Constrains (8) and (9) ensures all customers requiring the delivery or/and

pickup of an empty or/and full container. Customers who require a full container

will always be served from the terminal (10), while constrain (11) ensures the

truck must return to the terminal after a pickup of a full container.

Constraints (12) is a necessary indicator which turns one when a customer is visited and used in constrain (13) to link drivers with trucks to trips when visiting customers during a trip.(14) is used as an indicator to check if the same driver will drive the next trip.

The next constraints are for tracking the driving time and processing time at the customer node and start/ending node, also the timely arrival at time-windows is handled in this part. Constrain (15) make sure that the starting time for the first customer is earlier than the starting time of the next customer, note that the rest period is taken into account. While constraint (16) makes sure the truck does not wait at the terminal before the openings time of the time window, while constraint (17) makes sure a truck does not wait until the closing-time of the time window at the customer location and returns to the terminal. (18) takes care of a timely arrival between the specified time-window at the customer. Constraint (19) is necessary to define the start and ending time of the service at a customer. (20) Is use full to define the starting time if a driver takes multiple trips, note that the break is taken into account when the drivers rests at the terminal.

The next set of constraints are necessary to track the accumulative driving time

for each trip driven by driver d. Constrain (21) ensure the first trip for a driver will

start at time 0. (22) is necessary to set the driving time when the driver stops his

trip,and must be equal or less than the ending time for that trip. (23) is to set

the driving time to 0 for drivers not assigned to a trip. (24) is used to track the

driving time when visiting customers note that when the driver takes a rest the

driving time is reduced by α. Further (25) makes sure the driving time can never

be less than 0 when a driver takes a rest and the driving time is reduced by α,

in the previous constraint. The next constraint (26) will take care of the driving

time when the driver performs another trip, so the driving time of the next rout

is greater than the driving time of the previous trip. (27) makes sure a driver can

only rest once during the shift , while constrain (28) makes sure the continuous

driving time will never exceed 4,5 hours. Constraint (29-30) indicates the times

are always positive, and constrain (31 - 34) are the binary decision variables.

5 Numerical Experiments & Results

In the following section the results of the proposed model are shown, the first part explains how the data is generated and how the model should be used. In the second part the numerical experiments are explained and the results are shown.

5.1 Data generation

The data for the algorithm is generated as follows:

1. The type of customer is randomly generated the probability for the demand is equal for each customer, this means a customer can demand a pick up and/or delivery of an empty or full container.

2. The service time at a customer for a delivery or pickup is randomly distributed between 10 and 20 minutes, during this time the truck has to unload or load a container. When there is a delivery and a pickup the service time is randomly distributed between 20 and 30 minutes.

3. The travel time between customers is randomly distributed between 12 and

30 minutes. The randomly generated travel time is based on the geographical

dispersion of container terminals in the Netherlands, this can be seen in figure 5.1.

Figure 5.1: Overview geographical dispersion container terminals

4. There are four lengths of hard-time windows considered these are gener- ated by a discrete distribution which is based on the findings of Wang and Regan (2002), the length ranges from 5 minutes to 5 hours. The distribution can be found in table 5.1.

Length time window Probability

0.5 hour 0.1

2 hour 0.15

4 hour 0.35

5 hour 0.4

Table 5.1: Length of time windows by Wang and Regan (2002)

5.2 Input and output of the model

The following subsection will give information how the data is entered and read by the model, first we start to create a two-dimensional table with the travel time for each customer where the travel time for customer 0 = n + 1, these rep- resent the terminal. Each row from the two-dimensional table is than put behind each-other such that it forms one row, from which the model reads the travel time between the customers.

Secondly the time-windows are created for each customer, the terminal does not have a openings- and closing -time and assumed to be open at the start and end of each trip.

The service time for each customer is entered into a column starting from 0 to n + 1, in this model the service time for customer 0 and n+1 is set on 15, in which containers can be loaded at the start and unloaded at the end of each trip.

In the fourth step the binary table is created as can be seen in table 5.2, for the terminal {0, n + 1} there is no request. The table contains five columns:

column 1 contains the number of each customer, column 2 contains the pick-up of a full container, column 3 contains the request for a pick-up of an empty container, column 4 contains the request for a delivery of a full container and the last column contains a request for the delivery of an empty container.

Customer PF container PE container DF container DE container

0 0 0 0 0

1 0 1 0 0

2 0 0 0 1

3 1 0 0 1

4 0 1 1 0

5 1 0 1 0

6 0 0 0 0

Table 5.2: Example of request for five customers

For the last step the model needs to know the available amount of drivers/trucks,

customers and trips. There should always be enough drivers/trucks and trips avail-

able otherwise the model will not be able to generate a feasible output.. The model

will calculate the amount of trucks, drivers and trips needed to serve all customers while minimizing the total travel time.

Output

When the model is executed the generated output would look like the following:

Driver 1 with truck 2 performs trip 1 starting at 253 and ending at 823 Trip 1: 0 → 4 → 2 → 1 → 3 → 6

Driver 1 takes a break after serving node 4 c Driver 4 with truck 1 performs trip 2 starting at 430 and ending at 643

Trip 2: 0 → 5 → 6

5.3 Experiments

In this subsection the experiments and the results will be showed, for all ex- periments the number of available trips is equal to the number of customers so:

|R = N |

Experiment 1

The first experiment contains nine instances, all the instances are different

from each other to by changing the the amount of customers, available amount

of trips and trucks with drivers. The aim of this experiment is to investigate the

sensitivity and the computational time of the algorithm. The problem which must

be solved by the algorithm is complex that is the reason that the instances are

small ranging from 5 to 25 customers. Customers can be visited in a time period

of 8 hours, and the time windows are distributed according to Wang and Regan

(2002). An overview of the changing parameters of the first experiment can be

found in table 5.3.

Instance Customers Driver/trucks Trips

1 5 5 5

2 10 5 10

3 10 6 10

4 15 6 15

5 15 7 15

6 20 6 20

7 20 7 20

8 25 6 25

9 25 7 25

Table 5.3: Parameters of experiment 1 Experiment 2

The second experiment contains five experiments, in these experiments the width of the time-windows is adjusted and the amount of available drivers with trucks is set on 10. An overview of the changing parameter of the first experiment can be found in table 5.4. The goal of experiment 2 is to check how the model behaves with the necessary amount of drivers and trucks to serve all customers when the width of the time-window is narrowed down.

Instance Time-window width Openings-time Closing-time Customers

1 5 hour 0 5 10

2 4 hour 0 4 10

3 1.5 hour 0 1,5 10

4 0.5 hour 0 5 10

Table 5.4: Parameters of experiment 2 Experiment 3

While processing the computational results the observation was made the drivers

had a low utilization. The third experiment will compare the effects when the ob-

jective function which only minimizes the total amount of travel time between

customers (equation 1) is adjusted to an objective function which minimize the

total amount of travel time in the first term and in the second term the amount of drivers is minimized. The objective function is adjusted by adding a penalty (ϑ) every time when a truck driver is used (dr _d ). The modified objective functions is written in equation (35) and constraint (36) is added to indicate if a new driver is used. Constraint (36) the binary indicator dr _d will turn one every time when a new driver is assigned to perform a trip.

X

ij∈A

X

r∈R

x ^r _ij τ _ij + X

d∈D

dr _d ϑ (35)

dr _d = max

r∈R p ^r _d , ∀d ∈ D (36)

The data for 10 customers in table 5.5 are used to test both objective functions (equation 1 and equation 35), the same randomly generated data is used from experiment one.

Instance Customers Drivers/trucks

1 5 10

2 10 10

3 15 10

Table 5.5: Parameters of experiment 3 Experiment 4

During the first experiment it was noticed the truck waits at customer i if the time-window of customer j was closed. To avoid waiting at customers the following constraint is added:

S _ri ^start + s _i + τ _ij − S _rj ^start ≥ (x ^ij _r − 1)M, ∀ij ∈ A, r ∈ R

As a result drivers will return to depot if the time-window for customer j is still

closed. The driver will perform a new trip to serve customers which time-windows

do not overlap. The same data from experiment 1 are used, with the parameter in

table 5.6 .

Instance Customers Driver/trucks Trips

1 5 5 5

2 10 6 10

3 15 6 15

4 20 6 20

5 25 6 25

Table 5.6: Parameters of experiment 4

5.4 Setting

To solve the instances IBM ILOG CPLEX Optimization Studio V12.10.0 is used. The program is run on a personal computer with an Intel Core ^{R TM} i7 6700K CPU @ 4.0 GHZ and 16 GB of DDR4 RAM memory. The algorithm is run for 2 hours or till the optimal solution is found.

5.5 Results

In the following section the results will be presented which are obtained by per- forming the experiments in CPLEX.

Results experiment 1

The results of the first experiment are presented in table 5.7. The The gap in this

teable represents the approximation of the optimal solution if the algorithm would

not find the optimal solution under 2 hours, CPLEX would give a feasible solution

and the gap between the optimal solution in a percentage. The last column shows

the run time before the optimal solution is reached.

Instance Customers Drivers/Trucks Gap Optimum Time

1 5 5 Optimal 290 00.01 minutes

2 10 6 Optimal 520 00.03 minutes

3 10 7 Optimal 520 00.03 minutes

4 15 6 Optimal 710 04.45 minutes

5 15 7 Optimal 710 03.32 minutes

6 20 6 Optimal 1050 06.22 minutes

7 20 7 Optimal 1050 05.37 minutes

8 25 6 Optimal 1270 37.55 minutes

9 25 7 Optimal 1270 20.30 minutes

Table 5.7: Results of experiment 1 Results experiment 2

The results of the second experiment are presented in table 5.8 and shows the effects of the time window length on the amount of trucks and drivers assigned.

The number of assigned drivers/trucks is presented in the last column.

Instance Time-Window Gap Solution Time Drivers/trucks

1 0-5 Optimal 510 00.05 minutes 2

2 0-4 Optimal 510 00.05 minutes 4

3 0-1.5 Optimal 530 00.14 minutes 4

4 0-0.5 Optimal 760 00.05 minutes 10

Table 5.8: Results of experiment 2 Results experiment 3

The results of experiment three shows the average driving time and the amount of

drivers assigned of the solution proposed by the model. The results of the original

objective function and the adjusted objective function can be found in table 5.9

and table 5.10 respectively.

Instance Customers Gap Time Drivers/trucks Avg. drive time

1 5 Optimal 0.01 minutes 2 02.36 hour

2 10 Optimal 01.24 minutes 4 03.34 hour

3 15 Optimal 06.53 minutes 5 03.35 hour

Table 5.9: Results of experiment 3.1 with equation (1)

To generate the results in table 5.10, a penalty (ϑ) with the value of 1 is used:

(ϑ=1).

Instance Customers Gap Time Drivers/trucks Avg. drive time

1 5 Optimal 02.13 minutes 2 06.79 hour

2 10 Optimal 02.57 minutes 2 06.48 hour

3 15 Optimal 16.37 minutes 3 05.92 hour

Table 5.10: Results of experiment 3.2 with objective function 35 Results experiment 4

Here the results of experiment 4 are presented when the constrain is added to prevent trucks to wait at customer i for the openings-time of customer j.

Instance Customers Drivers/Trucks Gap Optimum Time

1 5 5 Optimal 290 00.02 minutes

2 10 6 Optimal 570 00.51 minutes

3 15 6 Optimal 720 09.32 minutes

4 20 6 Optimal 1050 20.30 minutes

5 25 6 Optimal 1280 64.50 minutes

Table 5.11: Results of experiment 4

6 Discussion & Limitations

The model can handle: hard time-windows, multiple trips, the pickup and delivery

of different type of containers and pays respect to the European driving regula-

tions. The results from all the experiments are presented in the previous section are discussed in this section. Also the limitations of the model are discussed in the last part.

6.1 Discussion

The model is slow to generate the optimal solution when the amount of cus- tomers increases the time to solve the model also increases. This was expected due to the fact a VRP with time-windows is NP-Hard (Lenstra & Rinnooy Kan, 1979) the algorithm will always find the optimal solution when enough time is given. A VRPDP is hard to solve by a MIP formulation, this means only small instances can be solved optimally (Gribkovskaia & Laporte, 2008). The input of the amount of drivers is also affecting the computational speed of the proposed model, see table 5.7. When there are too much drivers act as an input for the problem the computational speed drastically increases, this due to the fact the so- lution pool to solve for the algorithm increases. An unexpected result was that also less drivers impacts the computational speed, this can be explained because the algorithm has to combine trips and check all the time-windows to obtain the optimum. Overall with the presence of multiple-trips, time-windows and driving regulations the exact algorithm performs quite well and can generate an optimal solution with small instances. In a practical setting, the model could support the daily scheduling activities. The computational speed for bigger data sets can im- proved by using a meta-heuristic, this corresponds to the conclusion of Hernandez et al. (2014). Whereas exact algorithms will always obtain the optimal solution, meta-heuristics will obtain a near optimal solution in a reasonable time. Meta- Heuristics which are proposed to solve combinatorial optimization problems are:

Genetic Algorithm (GA), Tabu Search (TS), Simulated Annealing (SA), Parti- cle Swarm Optimization (PSO) and Ant Colony Optimization (ACO) (Ashouri &

Yousefikhoshbakht, 2017; Chen & Ting, 2006).

Time-windows

The time-windows in the model are hard, this means that the vehicle is not al- lowed to arrive earlier or later.

Experiment 2

During experiment 2 the model behaves as expected, if the length of the time win- dow decreases and the openings and closing times of the time windows are narrow the model will require more drivers to satisfy all customer demand. On the other hand if the closing and openings time of all customers are wider less drivers are required and the model will generate a feasible solution faster this can be seen in table 5.8. Experiment 4

As expected the model allows the driver to wait at customer i, if the time-window is closed at customer j. This only happens when customer i can satisfy the de- mand of customer j and if the travel-time to the terminal is longer than traveling to customer j. Constraining this behaviour had a significant impact on the computa- tional time, if necessary the following constrain can be added (Note that applying this constrain will substitute constrain 16 and 17):

S _ri ^start + s _i + τ _ij − S _rj ^start ≥ (x ^ij _r − 1)M, ∀ij ∈ A, r ∈ R

However in bigger instances and this behaviour is observed it is recommended to stretch the time-window at the particular customer i and/or j and create an over- lap. One could also argue in practice it does not matter if the driver wait at the customer or at the terminal, since the driver has to be paid either way. When the driver has to perform another trip instead of waiting at customer i extra travel time is added. This causes the optimal solution is shifted upwards, the difference can be noticed between experiment 1 (table 5.7) and experiment 2 (table 5.11)

In contrast to experiment 4 if the time-windows are relaxed (constraint 16 and 17) the model can achieve a better computational speed and solve an instance of 25 customers in 13.1 minutes with 7 drivers. However a truck will always start at time 0 and will wait at the customer location till the opening of the time window.

In this case the driver will still be able to work 9 hours however there is a lot of waiting time at customer locations, which seems unrealistic from a practical per- spective.

Increase average driving time

The model will minimize the total time travelled and does not minimize the amount

of drivers used for the optimal solution. This is due to the fact that the amount

of drivers is not penalized in the proposed model. The explanation for this phe- nomenon is that the algorithm will always search for the optimal solution with the minimal amount of travel time and does not take the amount of drivers into account. This explains the results in table 5.9 where the sum of drivers have a low average driving time. However, in a practical setting it might be more bene- ficial to have less drivers compared to have less travel time. This could modified by adding a penalty when an extra driver, this penalty is added by adjusting the objective function to the following:

X

ij∈A

X

r∈R

x ^r _ij c _ij + X

d∈D

dr _d ϑ

ϑ: Resembles a fixed cost to penalize the use of a new driver and truck.

dr _d : Is a binary decision variable and takes the value 1 when a driver is used.

c _ij : Represent a cost for travelling between customer i and j

Applying the adjusted objective function means the amount of drivers needed is decreased and as a result the average amount of driving hours increases. When applying the adjusted objective function (35) the travel time (τ ij ) should represent a variable cost of travelling between customers (c ij ). The penalization (ϑ) should correspond with a fixed cost every-time a new truck and driver is hired to perform a trip. The downside of the adjusted objective function is the increment of the computational time this can be seen in table 5.10.

6.2 Limitations

The model has several limitations. One of the limitations of the model is that not all issues are considered when the model is used in a practical setting. Such the fact drivers wait at customer location i if the time window is closed at customer j.

In practice a driver can also consider to keep a break during this waiting period or

perform an extra trip, this is not considered in the model. Also no costs are consid-

ered in the model and it is arguable if the driver should wait for the opening of the

time-window or the driver should return to the terminal. Returning to the terminal

could mean an extra handling for unloading/loading a container and travel time

is added. The next limitation in our problem the time-windows are opened for 8 hours, the closing time of the terminal is not considered in the problem. Another limitation is the a planning horizon of one day ranging from 6am till 10pm. If the planning horizon is stretched the modelled driving regulations do not cover all the European driving regulations.The last limitation is that the research is conducted under time pressure, therefore not all directions are thoroughly researched.

7 Conclusion & Future Research

In the last section a conclusion of the model is written and will end with di- rections for future research.

7.1 Conclusion

A model is constructed which considers the movements of full and empty con- tainers between customers and the terminal and aims to decrease the total travel time to serve all customers, container handling and empty trips. The addition to the existing literature involving multi trips is the combined pickup and delivery 1-M-1 for full containers and 1-M-M for empty containers and European driving added. Exact formulations which involve multi-trips are scarce according to Her- nandez et al. (2014), during their research only three papers are found involving exact formulations to solve a VRP with multi-trips. An exact formulation can be use-full in situations where a precise output is required. The addition of having an exact method instead of heuristic is the fact that heuristics might get stuck in a sub-optimal solution while exact methods will in general generate results with less errors (Mucherino, Liberti, Lavor, & Maculan, 2009).

The model can be used for management purposes to decrease the travel time to serve customers. The model aims to decrease the distances in which the truck travels empty by merging trips as this is done in Caris and Janssens (2009).

The model always tried to find the optimal solution in terms of travelling time

and does not take the average driving time into account, this can be solved by

adding the binary variable (dr _d ) and a parameter (ϑ) to the objective function and

add constraint 36. If the adjusted cost function is used, the model will try to find

the optimal travel time with the least amount of truck drivers.

7.2 Future research

For future research it would be interesting to improve the computational speed of the model by using ant colony optimization algorithm (ACO). . ACO is a meta- heuristic and searches the for the fastest arc by releasing pheromone each time a vehicle travels the same arc the same way as ants do. According to Xiong, Wu, and Wu (2017) ACO can easily be added to existing models. ACO was used to solve a Closed Loop Supply Chain model, the ACO outperformed MILP (Cplex) in terms of computational speed by 40% and obtained the global optimal solution for 98% (Esmaeilikia et al., 2016). The use of ACO is useful in VRP’s since the computational speed increases and the ability exists to use real-time information for the model (e.g. if there is last minute demand or a road block), this would increase the accuracy of the model in a real-life setting (Schyns, 2015).

Another addition to the proposed model would be to extend the planning hori- zon for the proposed model to one week. This implies that the driving regulations must be extended and the eleven hours of rest have to be added after each shift.

When the terminal also operates during the night, the drivers need a rest of 45 min-

utes after a continuous drive of three hours. The last addition of the model would

be the addition of different cargo such as hazardous cargo, in this case the model

needs to deal with a heterogeneous pool of drivers and containers. To haul specific

types of hazardous cargo a driver must be qualified according to REGULATION

(EC) No 68/2008 VolI (United Nations Economic Commission for Europe, 2019)

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