A Multi-Trip Model for Delivery and Picking Up
Containers with Driving Regulations
Supervisor: dr. ir. S. Fazi Second Supervisor: dr. I. Bakir
By
Marc Baër
Faculty of Economics and Business
Msc Technology and Operations Management
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Abstract
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Acknowledgement
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Table of Content
1. Introduction ... 5 2. Literature Background ... 6 3. Problem Description ... 9 4. Mathematical Model ... 10 4.1 Model Formulation ... 10 4.2 The Model ... 124.3 Discussion of the Model ... 14
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1. Introduction
Globally, transportation of containers plays an increasingly important role in the movement of goods worldwide. Within the container supply chain, inland transportation generates relevant cost and several drawbacks, especially due to excessive trucking. In fact, truck transport is still the main means of transport to move containers within the hinterland. In 2018, 44% of all freight weight in the Netherlands was transported by trucks and they carried 2,6% more goods than in 2017 (CBS, 2019).
Generally, the transportation of containers via truck consists of two activities: delivery and pick-up. Typically, trucks depart from a depot or terminal where containers are available. Each truck delivers one container to a certain destination and on the way back may pick up at the same destination or at another location another container to be transported at the initial terminal. Once these containers are delivered they will be further moved by other trucks or means of transport or they may be transported between terminals or further inland by other means of transport, but this is not in the scope of our research. See Figure 1 for a graphical representation, in which the dotted lines are not part of the scope of this research. The scope of our research is thus the delivery and picking up of containers from and to one central terminal by trucks and drivers.
One of the biggest challenges in this truck transport is allocating containers to trucks, trucks to rides, and rides to drivers. The planning has to be devised such that containers are delivered on time, trucks and drivers are used once in every time frame, and with goal of minimizing routing costs. Additionally, there are other constraints which are relevant in this setting, such as the amount of working hours for a driver.
The literature lacks models which consider the two activities of container transportation, along with deadlines for these activities and which considers drivers and their regulations into the model. However, there are several papers about driving regulations and routing scheduling (e.g., Goel & Gruhn, 2006; Goel, 2009; Prescott-Gagnon, Desaulniers, Drexl, & Rousseau, 2010). An important reason to study driving regulations in models is that they increase safety and punctuality in road freight transport, and the schedule of the drivers must
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apply to these regulations from the European Union. These papers do not consider multi-trip, thus a driver only is assigned to just one trip and not more than that trip. To make our model more real-life, we consider multi-trips in our variant (e.g., Hernandez, Feillet, Giroudeau & Naud, 2014; Mingozzi, Roberti & Toth, 2013; Brandao & Mercer, 1998).
In this paper, we tackle this important problem, which we indicate as a variant of the Multi Trip Vehicle Routing Problem with Backhauls (MTVRPB), in which the customers are divided into two subsets. One set contains customers which needs a container to be delivered and one set where customers need a container to be picked up (Caceres-Cruz, Arias, Guimarans, Riera, & Juan, 2014). In this problem, the deliveries need to be done before picking up the quantities. The problem in this paper is as follows: assuming a homogeneous container type and size with different deadlines and a homogeneous fleet of trucks and drivers who have limited working time lengths, the objective is to find the optimal assignment of the drivers and trucks to trips in order to minimize the total distribution costs. The problem is focused on both the delivery and pick up activity of the container transportation.
Our aim is to develop a mathematical model and by means of an experimental framework to analyze solutions in order to generate practical insights. The model is solved via CPLEX, an industrial solver.
This paper is structured as follows. In section 2, the relevant papers which study similar problems are reviewed. In section 3, a description is given of the problem which is addressed in this paper. In section 4, the methodology that we use is described. In section 5, the model is presented and in section 6 contains the results. Section 7 is for the discussion, limitations and future research directions. Finally, section 8 contains the conclusion.
2. Literature Background
The vehicle routing problem (VRP) has many variants. In this section, the different variants of VRP which are relevant and similar to our problem are discussed. The paper of Eksioglu, Vural and Reisman (2009) gives a review on the different variants of the VRP. Eksioglu et al. (2009) state that the literature about VRP is an ever-emerging field and compasses lots of variants.
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destination is introduced. Kolen, Kan & Trienekens (1987) add to this problem time windows in which each client needs to be served and describe a branch-and-bound method to minimize the total route length. In the paper of Dumas, Desrosiers & Soumis (1990), they tackle the problem with time windows, but add delivery and picking up activities, (VRPPD) which is capable of handling multiple depots and different types of vehicles and for which they propose an algorithm. The paper of Savelsbergh & Sol (1995) discuss why VRPPD is different than other variants of the routing problem and present a survey of the different problem types and solution methods. The study of Goetschalckx & Jacobs-Blecha (1989) consider the VRPPD in which the deliveries need to be done before any pick up activity and for which they propose a two-phased solution methodology. Ong (2011) add to this variant multiple trips and study a new variant considering backhaul, multiple trips and time windows (VRPBMTTW). They use Ant Colony Optimization (ACO) as the problem solving technique and test the model by using random data and analyse parameter changing. Hernandez et al. (2014) consider a VRPBMTTW, but in which a trip has a limit duration. The problem studied in the paper of Vidovic, Radivojevic & Rakovic (2011) is closely related to the VRP with Backhaul (VRPB), but the trucks can either carry one 40-foot container or two 20-foot containers simultaneously, so the trucks can visit 2, 3 or even 4 nodes in one trip, a feature the trucks in this paper cannot do. In the paper of Nossack & Pesch (2013), a truck scheduling problem is addressed which involves multiple terminals and in which they do not consider drivers, whereas in this paper only one terminal and the drivers are considered.
There are papers which consider the simultaneous pickup and delivery of goods (e.g., Wang, Mu, Zhao & Sutherland, 2015; Min, 1989; Cheng & Wu, 2006; Catay, 2010), which is indicated as VRPSSPD. This variant is related to the problem of this paper, because it considers pickup and delivery of goods, but in the VRPSSD these two activities can happen simultaneously, whereas in our variant delivery has to happen before pickup. A variant of the VRPSSD is where the fleet of vehicles is heterogeneously (e.g., Avci & Topaloglu, 2016), whereas we consider a homogenous fleet. Another difference of that problem and our problem is that drivers and their driving regulations are included in our problem.
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Goel & Gruhn (2006) studies the problem of the driving regulations for drivers with a vehicle routing problem and considers itself the first work regarding EU regulations regarding drivers’ working hours. The study of Goel & Gruhn (2006) considers all the EU regulations, whereas, for simplicity reasons, we do not. The paper of Goel (2009) continues with the work of Goel & Gruhn (2006). The paper of Prescott-Gagnon et al. (2010) adds time windows to the vehicle routing problem with drivers regulations. All these studies differ from ours by having more the focus on the driving regulations and consider simple vehicle routing problems. The similarities and differences with the papers in this section and our paper is shown in the Table 1.
Table 1. Overview of related articles Authors and date Problem? Single
terminal? Homogeneous fleet of trucks? Multi- trip? Deadlines or time windows? Delivery and pickup?
Drivers and their regulations?
Orloff (1974) M-GRP No Yes No Neither No No Kolen et al.
(1987)
VRPTW Yes Yes No Time windows No No
Dumas et al. (1990)
VRPPD Yes Yes No Time windows Yes No
Goetschalckx & Jacobs-Blecha
(1989)
VRPPD Yes Yes No Neither Yes No
Ong (2011) VRPBMTTW Yes Yes No Time windows Yes No Hernandez et al.
(2014)
MTVRPTW Yes Yes Yes Time windows No No
Vidovic et al. (2011)
VRPB Yes Yes No Neither No No
Wang et al. (2015)
VRPSSD Yes Yes No Deadlines Yes No
Avci & Topaloglu, 2016
VRPSSD Yes No No Time windows Yes No
Goel & Gruhn (2006)
VRPDWH Yes Yes No Time windows No Yes
Goel (2009) VRPDWH Yes Yes No Time windows No Yes Prescott-Gagnon
et al. (2010)
VRPDWH Yes Yes No Time windows No Yes
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To have a good starting point, the paper of Hernandez et al. (2014) is the basis for our model. From all the papers reviewed, their model is the most similar to ours, but differs from the proposed model in this paper in multiple ways. First, their model is about visiting customers in order to provide a service which takes some time and this service must happen between certain time windows. The main focus of our model is to visit locations in order to either deliver or pick up a container before a certain deadline, from which we do not take any service time or loading and unloading time into account. Second, the model of Hernandez et al. (2014) takes the capacity of the trucks into account, and states that more than one job can be loaded in the truck, whereas in our model the truck either has a container that needs to be delivered or has been picked up, or is empty. Thus, our model only considers one location at a time and at most two locations in a trip. Third, we add drivers and a part of their driving regulations in the model, whereas in their model they do not consider any drivers at all.
3. Problem Description
We tackle the problem of planning a set of available containers to a set of trucks and drivers, considering an initial single terminal where the transportation originates. Containers in our problem have one size and can either be
delivery or pickup and have a deadline. Delivery containers are available at the terminal and have to be delivered to determined locations. On the other hand, pickup containers are available at determined locations and have to be picked up and be transported to the terminal. Trucks have a capacity of one container and thus can transport one container per time. There is a cost
involved based on how long the route they cover is. The trucks are all the same, because we consider only one type of container and thus the fleet of trucks is homogenous. The trucks do not have any availability constraint, thus the trucks are available all week. The drivers in our problem are all of the same type, but a driver has to obey to certain regulations, such as the amount of time they may drive uninterrupted before taking a break and the maximum driving time per week. A driver can drive each truck of the fleet. The aim of the problem is to allocate containers to trucks. Of course, a truck can make multiple trips. A trip can be considered in three ways. First, a trip can be the amount of time it takes a driver to drive a truck with a
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container from the terminal to a location and drive back without a container. Second, a trip can be the amount of time it takes from the terminal empty to a location to pick up a container and to drive back to the terminal. Last, a trip can be the time it takes from the terminal with a container to a location and then drives to another location to pick up a container before it heads back to the terminal, which is graphically shown in Figure 2.
The objective of the model is to find the optimal assignment of the drivers and trucks to containers in order to minimize the total distribution costs, i.e. minimize the hours driven and the resting periods that need to be taken.
Because the focus of our model is the amount of driven hours and the resting periods, our model considers the following assumptions. We only consider one type of container, however in reality there are many different types of containers. Drivers and trucks are homogeneous, however in reality this may not be the case, because a driver can drive different type of trucks with different drivers’ licenses. With the drivers being homogenous, the assumption is that their working speed and wages are the same. For simplicity reasons are the loading and unloading activities of the container to the truck not considered in this model.
4. Mathematical Model
We model the problem of transporting containers with trucks from and to a terminal by assigning trucks and drivers to the containers. The available information concerns the amount of containers that need to be delivered and picked up, their deadlines, the available drivers and trucks and how much hours the drivers have driven and the distances between the locations, the terminal and each other.
The aim is to minimize the amount of hours driven and the resting periods that need to be taken, while the deadlines of the containers need to be met.
4.1 Model Formulation
The mathematical model formulation is as follows. Consider a graph where there is a set of arcs and vectors, denoted 𝐴 and 𝑉 respectively. Next to that, consider a set of containers 𝐶, which either need to be delivered or picked up. Each container has a certain due date Bi
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same capacity of one container, which is further not specified in the model and is not adjustable. The set of trips is denoted as 𝐾, in which 𝐾 ≤ 𝑁, because a trip considers at least one container which needs to be delivered or picked up.
A driver has certain regulations to which it must obey, such that it may not drive uninterrupted for more than the parameter 𝛽, before it needs to take a break with a duration of 𝑅 and it may not drive for more than 𝐺 hours per week.
To characterize the assignment of the locations of the set of trucks to trips, we set the binary assignment variable 𝑥𝑖𝑗𝑘 with 𝑘 ∈ 𝐾 and 𝑖, 𝑗 ∈ 𝑁, which is 1 if locations 𝑖 and 𝑗 are visited
during trip 𝑘 and 0 if otherwise. The binary variable ℴ𝑘𝑢 with 𝑢 ∈ 𝑈 and 𝑘 ∈ 𝐾 is 1 if trip 𝑘 is been assigned to vehicle 𝑢 and 0 if otherwise. The binary variable 𝑦𝑘𝑙𝑢 with 𝑢 ∈ 𝑈 and 𝑘, 𝑙 ∈ 𝐾 equals 1 if trip 𝑙 is been done directly after trip 𝑘 if they are both assigned to truck 𝑢, and 0 if otherwise. Similarly, the binary variable 𝑠𝑘𝑙𝑖 indicates whether trip 𝑙 is been done directly after trip 𝑘 if they are both assigned to driver 𝑑 ∈ 𝐷 . The binary variable 𝛿𝑖𝑘 indicates if container 𝑖 is being delivered or picked up during trip 𝑘. The other decision variables which are not binary are 𝑆𝑖𝑘, 𝑑𝑘𝑠𝑡𝑎𝑟𝑡, 𝑑𝑘𝑏𝑎𝑐𝑘, ℎ𝑖𝑘𝑠𝑡𝑎𝑟𝑡, ℎ𝑖𝑘𝑏𝑎𝑐𝑘, 𝑟𝑒𝑠𝑡𝑓𝑟𝑖𝑘. 𝑆𝑖𝑘 denotes the time location 𝑖 is being reached
in trip 𝑘. 𝑑𝑘𝑠𝑡𝑎𝑟𝑡 and 𝑑𝑘𝑏𝑎𝑐𝑘 are respectively the starting and ending time of trip 𝑘, whereas
ℎ𝑖𝑘𝑠𝑡𝑎𝑟𝑡 and ℎ𝑖𝑘𝑏𝑎𝑐𝑘 are the starting and ending time for driver 𝑖 for trip 𝑘. The last decision
variable 𝑟𝑒𝑠𝑡𝑓𝑟𝑖𝑘 is for the amount of breaks driver 𝑖 needs to take during trip 𝑘. The sets,
parameters and the decision variables are summarized in Table 2.
Table 2. Summarizing of the sets, parameters and decision variables Sets
𝑪 The set of containers 𝑵 The set of locations 𝑼 The set of trucks 𝑫 The set of drivers 𝑲 The set of trips
Parameters
𝜷 The maximum uninterrupted driving hours 𝑹 The duration of a break
𝑮 The maximum driving hours per week
Decision variables
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𝒛𝒊𝒌 Binary indication if trip 𝑘is assigned to driver 𝑖
𝒚𝒌𝒍𝒖 Binary indication whether trip is 𝑙 done directly after trip 𝑘 by vehicle 𝑢 or not
𝒔𝒌𝒍𝒊 Binary indication whether trip is 𝑙 done directly after trip 𝑘 by driver 𝑑 or not 𝜹𝒊𝒌 Binary indication if container 𝑖 is being delivered or picked up during trip 𝑘
𝑺𝒊𝒌 Time location 𝑖 is reached in trip 𝑘
𝒅𝒌𝒔𝒕𝒂𝒓𝒕 Leaving time at depot for trip 𝑘
𝒅𝒌𝒃𝒂𝒄𝒌 Arrival time of trip 𝑘 back at depot 𝒉𝒊𝒌𝒔𝒕𝒂𝒓𝒕 Start time of trip r
k from driver 𝑖
𝒉𝒊𝒌𝒃𝒂𝒄𝒌 Time that driver 𝑖 returns at depot from trip 𝑘
𝒓𝒆𝒔𝒕𝒇𝒓𝒊𝒌 Amount of rest periods that driver 𝑖 needs to take during trip 𝑘
4.2 The Model
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Constraint (1.2) & (1.3) ensure that each container is delivered or picked up, depending on what type of activity is needed. Constraint (1.4) ensures that double trips are not possible, while (1.5) ensures that there is one trip at the time. Constraint (1.6) & (1.7) are constraints for the sequence of the trips, where (1.6) ensures that a truck goes straight from the terminal (0) to the delivery location for the container, while (1.7) makes sure that the truck goes from the pick-up location of a container to the terminal. Constraint (1.8) ensures that location j can be reached in time when it comes after i in trip rk. Constraint (1.9) is in in place to make sure
that the time back at the terminal is later than travel time plus the arrival time at the destination if that trip is driven. Constraint (1.10) makes sure that the arrival time at location i in trip rk is later than the start time plus the travel time if those are in the same trip. Constraint
(1.11) makes sure that each of the deadlines of the containers are met. Constraint (1.12) is about assigning trips to trucks, where constraint (1.18) is for assigning trips to drivers. Constraints (1.13) and (1.14) ensure that if trip rk and trip rl are driven by vehicle u, one of the
trips is done after the other, constraint (1.19) and (1.20) do the same for the drivers instead of trucks. Constraint (1.15) makes sure that if start time of trip rl is earlier than trip rk, then
these two trips cannot be scheduled after each other for the same truck, and (1.21) does the same for the drivers. Constraint (1.22) & (1.23) make the starting time of a trip the same as the starting time for the driver that is assigned to that trip, and (1.24) & (1.25) do the same for the time back at the terminal of a trip. Constraint (1.26) is for the amount of rest periods a driver needs to take on each trip. Constraint (1.27) ensures that the time a trip starts is earlier than the end time of a trip. Constraint (1.28) ensures the regulation of the maximum working hours per week is followed. Constraints (1.29) to (1.34) state that those decision variables are binary variables and thus only can take a value of 0 or 1.
4.3 Discussion of the Model
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5. The Experiments
In this section we show the results of the experiments we run on our model. The aim of these experiments is to see how the model reacts to different sizes of experiments, to check the behaviour of the model and to perform a sensitivity analysis. First, we discuss how the experiments are created. Second, we show the setting in which the experiments are run. As last, we show the results of the instances.
5.1 Instances generation
We generate 8 instances. The instances differ in the amount of containers that need to be delivered or picked up, the number of drivers and how many trucks there are available. Due to complexity, we create small instances from 10 to 25 containers. The overview of the instances is shown in table 1.
There are three parameters in the instances that are generated randomly. The first is the time it takes from location i to location j with hours between 1-10, where 𝑡𝑖𝑗
= 𝑡𝑗𝑖. The second is the deadlines of the
containers, whereby a deadline cannot be
later than hour 168, because the planning horizon of our model is one week and one week has 168 hours. As last, the specification of a container about whether it needs to be picked up or be delivered. These three things, the driving hours between locations, the deadlines and whether a container needs to be picked up or delivered, can be easily changed according to whatever dataset needs to be solved.
5.2 The Setting
In order to solve the instances, we use IBM ILOG CPLEX Optimization Studio, version 12.9.0, which is an industrial solver and commonly used to solve these kind of instances. We run this on a personal computer with an Intel(R) Core (TM) i5-8250U CPU @ 1.60 GHz and 8.00 GB RAM memory. The maximum time limit we give CPLEX to solve the instances is 30 minutes or the computation stops when the program runs out of memory.
5.3 The Results
Experiment 1 took 3 minutes and 25 seconds to find the optimal solution, whereas experiment took 4 minutes and 9 seconds for this. For each of the experiments 3 to 8, CPLEX cannot find the solution within the time limit. Therefore, the results are the best one found
Instance # containers # drivers # trucks
1 10 2 2 2 10 3 3 3 15 3 3 4 15 4 4 5 20 5 5 6 20 6 6 7 25 6 6 8 25 5 5
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by the program, but these is not the optimal solution. The objective for experiments with the same amount of containers are the same. The reason for this is that the model finds the best combination of locations it can combine to minimize the driving hours and thus has the least necessary resting periods. The objective is the same because it consists of the amount of driven hours and the resting periods that are taken.
Experiment Gap # containers # drivers # trucks
Best Integer Best node Time 1 Optimal solution 10 2 2 95 95 3.25 2 Optimal solution 10 3 3 95 95 4.09 3 12.26% 15 3 3 81 71,0657 30 4 11.11% 15 4 4 81 72,0010 30 5 12.58% 20 5 5 151 132,000 30 6 11.92% 20 6 6 151 133,000 30 7 13.02% 25 6 6 169 147,000 30 8 13.02% 25 5 5 169 147,000 30
Table 4. Results of the experiments with basic model
From the data that is listed in table 2, we can draw some conclusions. The model can solve small instances in very little time. When the instances become larger, CPLEX does not find the optimal solution in a maximum time of 30 minutes and the gap for these experiments is around 12%. These findings are in line with the fact that the model is NP-hard.
One thing that the model tends to do is to use all drivers that are available and assign them to trips. To investigate and change this behaviour of the model, we make some adjustments to the model. A result of this is, is that none of the drivers really gets close to the maximum driving hours. Also, it is not very logical to let all the available drivers drive, while all constraints can still be met with fewer drivers. For that reason, we add the binary variable 𝑢𝑖
to constraint (1.28), which is 1 if a driver is being used in the schedule, and is 0 otherwise. This is indicated in the constraint (1.35). Constraint (1.28) is now the following:
∑𝑘∈𝐾(ℎ𝑖𝑘𝑏𝑎𝑐𝑘− ℎ𝑖𝑘𝑠𝑡𝑎𝑟𝑡) ≤ 𝐺 ∗ 𝑢𝑖 , 𝑖 ∈ 𝐷 (1.28a)
𝑢𝑖 ∈ {0,1} (1.35)
Also, the objective function needs to be adjusted to minimize the amount of drivers that is being used per schedule. We add the variable 𝑢𝑖 and multiply this with a cost 𝑃 for
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function. In the first experiments, the weight of the resting periods and the driving hours were 50/50, but now this ratio can be changed by changing ∝. The objective function is now changed to:
min ∑𝑘∈𝐾∑(𝑖,𝑗)∈𝐴𝑡𝑖𝑗𝑥𝑖𝑗𝑘 + 𝑢𝑖∗ 𝑃+ ∝∗ ∑𝑖∈𝑈 ∑𝑘∈𝐾 𝑟𝑒𝑠𝑡𝑓𝑟𝑖𝑘 (1.1a)
The rest of the model stays the same and this change of the model is tested by solving an instance with 7 containers, 7 drivers and 7 trucks. We chose this amount of containers, because CPLEX is capable of solving this instances optimally in the time limit. We let 𝑃 and ∝ change to 1, 5, and 10, thus performing 9 experiments. The results of these experiments are shown below. First, the maximum time a driver is allowed to work, thus G, is 48, but we change this later to 40, 24 and 20 to see how this changes the results. These settings are chosen for the following reasons. When a person works full-time, they normally work 40 hours per week. It is also possible to work part-time, which is normally around 20-24 hours per week. The results are in the graphs below.
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In the tables below are the driving hours per experiment and how many drivers were used per experiment. In the first column, the values of 𝑃 and ∝ are described, in the second how much drivers are used in the schedule and in the last column are the amount of hours per driver. For example, in Table 4 in the experiment with 𝑃 = 10 and ∝ = 1, one driver is scheduled to drive for 37 hours and the other for 6 hours.
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Table 5. Amount of hours driven per driver for G = 48
Value of 𝑃 and ∝ Amount of drivers used Driver times per driver
𝑃 = 1 and ∝ = 1 1 48 𝑃 = 5 and ∝ = 1 1 48 𝑃 = 10 and ∝ = 1 1 47 𝑃 = 1 and ∝ = 5 1 43 𝑃 = 5 and ∝ = 5 1 43 𝑃 = 10 and ∝ = 5 1 48 𝑃 = 1 and ∝ = 10 1 41 𝑃 = 5 and ∝ = 10 1 45 𝑃 = 10 and ∝ = 10 1 43
Table 6. Amount of hours driven per driver for G = 40
Value of 𝑃 and ∝ Amount of drivers used Driver times per driver 𝑃 = 1 and ∝ = 1 2 1: 39, 2: 9 𝑃 = 5 and ∝ = 1 2 1: 27, 2: 20 𝑃 = 10 and ∝ = 1 2 1: 37, 2: 6 𝑃 = 1 and ∝ = 5 2 1: 40, 2: 2 𝑃 = 5 and ∝ = 5 2 1: 24, 2: 17 𝑃 = 10 and ∝ = 5 2 1: 24, 2: 17 𝑃 = 1 and ∝ = 10 2 1:23, 2: 22 𝑃 = 5 and ∝ = 10 2 1: 35, 2: 9 𝑃 = 10 and ∝ = 10 2 1: 27, 2: 22
Table 7. Amount of hours driven per driver for G = 24
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Table 8. Table 5. Amount of hours driven per driver for G = 20
Value of 𝑃 and ∝ Amount of drivers used Driver times per driver 𝑃 = 1 and ∝ = 1 3 1: 17, 2: 14, 3: 13 𝑃 = 5 and ∝ = 1 3 1: 17, 2: 14, 3: 13 𝑃 = 10 and ∝ = 1 3 1: 17, 2: 14, 3: 13 𝑃 = 1 and ∝ = 5 3 1: 20, 2: 20, 3: 6 𝑃 = 5 and ∝ = 5 3 1: 17, 2: 14, 3: 13 𝑃 = 10 and ∝ = 5 3 1: 17, 2: 14, 3: 13 𝑃 = 1 and ∝ = 10 3 1:20, 2: 19, 3: 5 𝑃 = 5 and ∝ = 10 3 1: 17, 2: 14, 3: 13 𝑃 = 10 and ∝ = 10 3 1: 17, 2: 14, 3: 13
7. Discussion
We did four different settings in which 𝐺differs from 48, 40, 24 and 20. With 𝐺 equals 48, all containers could be transported with just one driver and meet all deadlines. The lowest time driven is found when ∝ is the largest. A reason for this behaviour is that it only uses one driver and thus has not much to do with the cost for using a driver in the schedule. With ∝ larger, the resting periods weigh more heavily on the objective function and it tries to reduce these and thus finds shorter trips in order to keep the objective function as low as possible. The trendline in the graph shows that when ∝increases, the total time that is driven reduces.
When 𝐺equals 40, it is no longer achievable to use one driver and meet all constraints, thus two drivers are used in the schedule. With ∝ being small, the distribution of the working hours is fairly unfair, one driver works almost to the maximum, where the second driver may take the rest of the hours. Another option is that the model distributes the working hours along the two drivers and thus let them drive almost the same hours. These two options happen randomly and there is not a clear pattern for when which option is chosen. A cause may be the setting in which this was done, thus the distances between the locations and the determination which containers needs to be delivered and which needs to be picked up. The trendline in this graph is also very flat, thus from this graph we cannot draw any conclusions about the behaviour of ∝ and 𝑃.
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trips to make this possible. In the cases that 𝑃 equals 5, the total driven time is lower than when 𝑃 is either 1 or 10. This shows that a larger 𝑃 does not result in a lower total driven time. The trendline of this graph is also descending, which concludes that the larger ∝, the total time driven reduces.
When 𝐺 is 20, three drivers are used instead of two drivers by 𝐺 is 24. Almost for every setting of 𝑃 and ∝, the total time driven is the same, except for when 𝑃 is 1 and ∝ is 5. Along with the setting where𝑃 is 1 and ∝ is 10, these two settings have different distributions of the working hours among the drivers. With 𝐺 equals to 20, it shows that just by tweaking a parameter, the behaviour of the model can change. As can be seen in all graphs, except the graph where 𝐺 is 40, the trend lines are descending the larger ∝ becomes. Thus, in order to reduce the total amount driven, ∝ has to become large.
The model is also changeable for certain situations. We showed this by changing the amount of hours that a driver maximum may drive per week. The model is up to change to handle other situations as well, whereby the hours that a driver may drive before a break is shorter or the maximum hours a driver may work is shorter. For example, when there are 4 drivers who all have a contract in which they need to work a maximum of 20 hours per week, the parameter can change to this amount of hours.
As with any mathematical model, there are certain limitations to the model conducted in this paper. First, not all driving regulations are included in this model. Second, the model does not take any time into account for loading or unloading the container, whereas in real life this would be the case. Third, the model does not take any other cost into account except the cost for using a driver in the schedule. Fourth, this research is conducted under a time limit in which not all directions are researched. Directions for further research are testing the model with more instances and with a longer time limit for CPLEX and tweaking the parameters more to investigate the behaviour, adding more driving regulations into the model, or take into account the time it takes to load or unload a container to make the model more like real life scenarios. An interesting research direction is to consider a heterogeneous set of drivers instead of a homogeneous set.
8. Conclusions
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set of trucks and driving regulations. This model is tested by performing experiments with different amount of containers and drivers and it shows that the model is capable of solving small instances, but it starts to struggle to find the optimal solution when the instances become larger. After that, the behaviour of the model is analysed and shows that the model uses all available drivers, even when it is not necessary. By adding the binary variable 𝑢𝑖 and
multiplies this with a cost function 𝑃, and adding a weight to the resting period in the objective function, the model does not show that behaviour anymore. It is also shown that the behaviour of the model can be changed with changing the cost function and the weight in the objective function to be used in more situations.
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