An optimization model for detailed scheduling of heterogeneous fleet of log trucks considering synchronization

ABSTRACT

Log transportation accounts for a significant portion of the total delivered cost of logs due to the considerable number of truckloads between origins and destinations. Hence, an efficient transportation plan at the operational level can generate cost savings for forest companies. In this paper, a mixed integer linear programming model is developed for daily routing and scheduling of heterogeneous trucks considering synchronization constraints. Continuous time representation is used for modeling the problem to generate accurate schedules and to synchronize the trucks and loaders at cut blocks and sort yards. Compatibility requirements, overtime, and decisions related to the trucking contractors are incorporated in the model. The outputs of the model include the number of trucks utilized by each contractor, the arrival times and waiting times of trucks at each location, the detailed schedule of loaders, and the amount of overtime assigned to the drivers. To validate the model, it is applied to test problems from a case of a Canadian forest company, where transportation activities are contracted out. Results show it is more economical to pay overtime than dispatching additional trucks to carry logs. Additionally, variable costs and maximum driving time have the most impact on the total transportation cost.

KEYWORDS:

• Mixed integer programming
• log transportation
• truck routing

Introduction

In many regions around the world, trucks are used to transport logs from forest areas to yards, mills, and other industrial sites. Due to the high volume of supply/demand and the limited capacity of trucks, a large number of truckloads are required to meet the demand. In 2014, almost 1.4 million truckloads were transported in British Columbia, Canada (British Columbia Forest Safety Council 2015). Therefore, as expected, transportation cost accounts for a considerable portion of log procurement costs. In some countries such as Chile, delivering logs to mills can contribute more than 45% of the operational costs (Audy et al. 2012). Hence, improvements in transportation planning could potentially result in cost savings for the forest industry (Palmgren et al. 2004).

Log transportation planning includes decisions at different levels. While long-term and medium-term decisions are made at strategic and tactical levels, operational plans are mainly related to short-term decisions such as routing and scheduling of log trucks (Malladi and Sowlati 2017). Routing of a log truck involves determination of the sequence of locations that the truck should visit, while times are assigned to each transportation activity in the scheduling of the log trucks. The fleet of log trucks can be either homogeneous or heterogeneous. Homogeneous fleets consist of the same truck type, while heterogeneous fleets consist of more than one truck type with different characteristics such as capacity and costs. In the routing and scheduling of log trucks, some practical aspects have to be considered in the operational transportation planning, for example, the operational time of a mill/yard during which the logs should be delivered and the maximum driving time of a truck driver. Furthermore, there are resources that should operate simultaneously to process a transportation task. Log loaders and trucks, for example, work together for loading and unloading of truckloads. By synchronizing log trucks and loaders, waiting times can be reduced, resulting in more efficient transportation planning.

Due to the importance of transportation, previous studies proposed different methods to improve and optimize log transportation at the operational level. In a group of studies, the flow of logs between different supply and demand points was considered as decision variables of models. In other words, these flows were outputs of the proposed optimization models. This group of studies is referred to log truck scheduling problems (LTSP) in the literature (Malladi and Sowlati 2017). In this group, some papers considered a daily planning horizon and determined flows of logs, routing and/or scheduling decisions. There are also other studies considering LTSP with a longer planning horizon. In these papers, different periods are linked to each other using inventory variables. Flisberg et al. (2009) considered LTSP for a 1-week planning horizon. They addressed the problem in two phases, in which the first phase determined daily flows of logs using a linear programming model. The second phase employed a Tabu Search to obtain near optimal solutions. Their method could perform well for different Swedish forest companies.

In another group of papers, the flow of logs was either known and used as inputs or was driven from an upper-level model. As Malladi and Sowlati (2017) mentioned, this group is referred to as a timber transportation vehicle routing problem (TTVRP). In TTVRPs, the planning horizon is 1 day, and the main outputs of models are routing and/or scheduling of log trucks. Gronalt and Hirsch (2007) introduced a Tabu Search-based solution approach for TTVRP to minimize the empty travel time of heterogeneous trucks dispatched from multiple home bases. Their proposed approach was applied to numerical examples and could generate good solutions. Daily routing and scheduling of homogeneous fleet of trucks with foldable containers was studied by Zazgornik et al. (2012). The authors proposed a mixed integer programming (MIP) model and Tabu Search for solving the problem. The performance of the algorithm was assessed by small, medium, and large-sized instances. Haridass et al. (2014) provided a non-exact solution method to address daily routing and scheduling of an American company and could reduce unloaded travel distance by 28%.

Table 1 summarizes the papers in the log transportation scheduling and routing literature. As it can be observed, both homogeneous and heterogeneous fleets have been considered in routing and scheduling of log trucks in the literature. The proposed models were applied to case studies or test problems. Synchronization of machines, however, did not receive sufficient attention in log transportation problems where the trucks do not have self-loading equipment.

Table 1. Summary of the log transportation scheduling and routing literature.

Problem type	Truck type	Papers	Case study/test problem	Routing/Scheduling	Synchronization
Log truck scheduling problem with a 1-day planning horizon	Homogeneous trucks	Palmgren et al. (2003)	Case study	Scheduling	No
		Murphy (2003)	Case study	Routing	No
		Palmgren et al. (2004)	Case study	Scheduling	No
		Bordón et al. (2018)	Case study	Routing	No
		Vitale et al. (2021)	Case study	Routing	No
		Melchiori et al. (2022)	Test problem	Scheduling	Yes
	Heterogeneous trucks	Rey et al. (2009)	Test problem	Scheduling	No
Log truck scheduling problem with a longer planning horizon	Homogeneous trucks	El Hachemi et al. (2013)	Case study	Scheduling	Yes
		El Hachemi et al. (2015)	Case study	Scheduling	Yes
	Heterogeneous trucks	Flisberg et al. (2009)	Case study	Scheduling	No
		Rix et al. (2015)	Test problem	Scheduling	Yes
		Ghotb et al. (2022)	Case study	Scheduling	No
Timber transportation vehicle routing problem	Homogeneous trucks	El Hachemi et al. (2011)	Case study	Scheduling	Yes
		Zazgornik et al. (2012)	Test problem	Scheduling	No
		Oberscheider et al. (2013)	Test problem	Scheduling	No
		Haridass et al. (2014)	Case study	Scheduling	No
	Heterogeneous trucks	Gronalt and Hirsch (2007)	Test problem	Scheduling	No
		Hirsch (2011)	Test problem	Routing	No
		Derigs et al. (2012)	Test problem	Routing	No

Transportation planning is also investigated for other forest products including biomass found in studies conducted by Han and Murphy (2012), Malladi et al. (2018), and Soares et al. (2019). Han and Murphy (2012) introduced a mathematical programming model for routing of heterogeneous trucks carrying predetermined truckloads of woody biomass in western Oregon. Malladi et al. (2018) considered biomass transportation planning for a one-week planning horizon. The authors employed a decomposition approach where daily flows between supply and demand points were determined through an MIP model in the first phase. An integer programming (IP) model was developed in the second phase to obtain routing decisions. Soares et al. (2019) addressed routing and scheduling decisions for homogeneous trucks by presenting an MIP model. The authors considered multiple synchronization constraints for different machines. In their problem, lorries carrying the loaders needed to be synchronized with the loaders. Also log trucks and loaders had to be synchronized for loading the trucks. In the furniture industry, Coelho et al. (2016) and Darvish et al. (2016) investigated transportation planning. Coelho et al. (2016) suggested an MIP model for routing of trucks considering time windows and lunch breaks for drivers. Darvish et al. (2016) presented an IP model for a one-year planning horizon. The model could specify daily flows that need to be transported. However, scheduling decisions were not included in their model.

Log transportation at the operational level has evolved considerably in recent years. In the literature, different aspects such as multiple depots (e.g. Flisberg et al. 2009), weight requirements (e.g. Gronalt and Hirsch 2007) and time windows (e.g. Oberscheider et al. 2013; Haridass et al. 2014) were addressed for routing and scheduling of log trucks. Further studies, however, are still needed to make the transportation plans more practical. According to a study by Lagzi (2016), in the scheduling optimization problems, the representation of time significantly affects the quality of solutions. Time can be represented by finite discrete intervals or continuous variables. In finite discrete representation, activities are assigned to finite intervals and schedules are generated accordingly, while in continuous time representation, the schedule of activities is defined by minutes or hours. Both discrete and continuous time representations have strengths and limitations. It is easier to use discrete intervals for modeling the scheduling problem for shared resources such as loaders used for loading/unloading activities (Lee and Maravelias 2018). The starting time of each activity of the resources is assigned to the beginning of discrete intervals. However, defining the approximated intervals is challenging and affects the accuracy of the schedules. In the continuous time representation, the schedules are more accurate because activities can happen at any time in the planning horizon. Moreover, variable processing times can easily be addressed by continuous variables (Lagzi 2016). However, modeling the problem for shared resources is more difficult using continuous variables. Despite all aforementioned advantages and drawbacks, there is no general consensus about time representation in scheduling optimization problems (Stefansson et al. 2011). Most papers in the log transportation literature (e.g. El Hachemi et al. 2013; Melchiori et al. 2022) used discrete time representation, while employing continuous variables in log transportation scheduling did not get enough attention.

Having detailed and accurate schedules is more important in synchronization of log loaders and trucks to calculate waiting times for loading/unloading because inaccurate waiting times may violate time windows (i.e. operational time of a mill) and impact the efficiency of the entire schedule. However, only a few papers addressed synchronization of log loaders and trucks and determined detailed schedules. El Hachemi et al. (2011), El Hachemi et al. (2013), El Hachemi et al. (2015), and Melchiori et al. (2022) used discrete time slots to facilitate synchronization of log loaders and homogeneous trucks. However, defining time slots can be more challenging when there is more than one type of truck and loading/unloading times depend on the type of the truck. Additionally, to synchronize log loaders and trucks efficiently, the discrete intervals need to be granulated enough which results in more binary and integer variables and makes the problem more difficult to solve compared with the continuous time variables. Soares et al. (2019) used continuous times in synchronization of different machines including trucks, loaders, and lorries in the biomass transportation problem. However, they had the synchronization only at the pickup points, where loaders with lorries and trucks with loaders needed to be synchronized.

In addition, real-world complexities of operations have usually been disregarded in the log transportation literature. Incorporating these complexities helps the transportation plans become more accurate and practical. Trucks, sometimes, cannot carry specific truckloads or visit certain locations. Therefore, incompatibility of trucks and their loads in the scheduling problem should be avoided. Also, it is noteworthy that in practice, forest companies can contract out some of their transportation tasks to trucking contractors for delivering the truckloads. As a result, these contractors and their limitations need to be incorporated into the operational plans. Only a few papers addressed compatibility requirements and contractual mandates in the forest products transportation literature. Malladi et al. (2018) considered compatibility requirements in routing of trucks carrying forest-based biomass. Ghotb et al. (2022) incorporated both compatibility requirements of trucks and trucking contractors in the determination of daily flows of log trucks. However, to the best of our knowledge, compatibility requirements and contractual mandates have not been addressed in the detailed scheduling of log trucks.

Moreover, truck drivers can be assigned overtime by the trucking contractors to fulfill the transport tasks. Working overtime usually has a higher cost rate compared to the regular working time. Depending on the contract, there is a maximum limit for assigning overtime for each truck driver. Incorporating overtime in log transportation planning is useful for decision makers as it gives them an understanding of the trade-off between assigning overtime and dispatching a new truck to deliver the truckloads.

To address the identified gaps, in this paper, an MIP model is developed to determine the detailed scheduling of heterogeneous trucks by employing continuous variables representing the schedule times. The proposed model considers trucking contractors, synchronization of trucks and loaders at each location, compatibility requirements, and overtime. The performance of the model is investigated using test problems from a real case of a large forest company in British Columbia, Canada.

Materials and methods

Case study

In this research, the case of log transportation for a large forest company in British Columbia, Canada, is investigated. The company predominantly manages Crown tenures with annual harvesting of roughly 6 million m³ of logs for its mills’ consumption and other external markets. The forest company procures different species such as western hemlock (Tsuga heterophylla) and Douglas fir (Pseudotsuga menziesii) with different log diameter classes. As the mills have different demands for logs of different species and diameter classes, the harvested logs at cut blocks need to be transported to sort yards for further processing before delivering them to the mills (demand points). Logs at the roadside of the cut blocks are either sorted or unsorted. Sorted truckloads require no further sorting at the sort yards. The processing activities at the sort yards include scaling, sorting, bundling, and dumping logs into the water. Due to the company’s request, we focused on the transportation planning in one of their operational regions where truckloads are sent to three sort yards for processing.

The fleet of trucks delivering the logs to the sort yards are heterogeneous. These trucks are either highway or off-highway trucks. Due to transportation regulations, off-highway trucks are not allowed on highways, while highway trucks can travel on all roads. The highway trucks have a lower unit transportation cost per time. However, off-highway trucks have a higher volume capacity compared to highway trucks, resulting in a lower cost per volume compared to that of a highway truck. Thus, there is a potential trade-off between dispatching different types of trucks. As shown in Table 2, it takes more time to load an off-highway truck. However, the unloading time for both highway and off-highway trucks is the same and is done by a machine called wagner at sort yards. Moreover, there are important compatibility requirements that should be considered in the assignment of truckloads to the trucks. Highway trucks can deliver both sorted and unsorted truckloads, while off-highway trucks can only transport unsorted logs. Sorted logs are bundled on the trucks by a bandit machine and lifted by a grapple at the sort yards. Off-highway trucks cannot carry sorted logs because the capacity of logs in these trucks is more than the capacity of grapples.

Table 2. The capacity, loading and unloading times, and compatibility requirements of trucks.

Truck types	Capacity (m^3)	Loading time (min.)	Unloading time (min.)	Compatibility
Highway trucks	43	15	8	Sorted and unsorted logs
Off-highway trucks	95	20	8	Unsorted logs

In the study area region, there is a central depot from which all trucks start their work and return to it at the end of the day. The depot is close to one of the three sort yards. Table 3 summarizes the characteristics of the sort yards. The second and the third columns of Table 3 represent the traveling time between the sort yards and the depot for the highway and off-highway trucks, respectively. The traveling time is for a one-way trip without considering loading and unloading times. The central depot is very close to sort yard 3 so the traveling time is zero. The fourth column of Table 3 shows the number of loaders at each sort yard. The space and processing capacities of the sort yards are listed by Ghotb et al. (2022).

Table 3. Number of loaders in each yard and traveling times of trucks to yards.

Sort yards	Traveling time to/from depot (min)		No. of loaders
	Highway trucks	Off-highway trucks
Yard 1	120	144	1
Yard 2	126	132	1
Yard 3	0	0	1

The forest company has contracted out transportation activities to trucking contractors. Each contractor has a specific number of trucks, and they deliver the truckloads from certain cut blocks based on their contract with the forest company. These contractors pay their drivers based on the driving time. A regular shift is for 8.5 h. In addition, if a truck needs more time to deliver its assigned truckloads, the contractors assign up to 2.5 h of overtime to the truck driver.

In our previous paper (Ghotb et al. 2022), we developed a mathematical model to determine daily truckloads between cut blocks and sort yards while balancing contractors’ workloads, and considering the processing and space capacities of the sort yards. Therefore, each truckload was identified by a pickup (cut block) and delivery (sort yard) location, truck type, and responsible contractor. The trucking contractors dispatch their trucks to deliver their assigned truckloads to the sort yards. As there is one loader at each sort yard to unload truckloads, it is important to have a transportation plan that can integrate truckloads of all contractors. This can avoid waiting times and reduce transportation costs for the contractors. Now, in this paper, we aim to have an optimization model to help contractors assign the truckloads to compatible trucks and determine a detailed schedule of each activity on each day. Currently, no optimization tool is used by the company for detailed scheduling of transportation activities. This research aims to address the problem and proposes an MIP model to synchronize log trucks and loaders and to determine the detailed schedules and waiting times with minimum total time.

The number of truckloads between each pair of cut blocks and sort yards is determined by an upper-level model, developed by Ghotb et al. (2022). Hence, the pickup and delivery locations, the responsible contractor, and the truck type for each truckload are the inputs of the current model. The truck drivers can have more than one trip per day, and they can be assigned overtime to deliver their truckload(s). It is assumed that there is a loader at each cut block and sort yard for loading and unloading activities. The loaders and trucks need to be synchronized. It means that when a truck arrives at a location, it has to wait until the loader becomes idle.

Network representation

The scheduling problem is defined on a network where $\mathit{P}$ and $\mathit{D}$ are pickup (cut blocks) and delivery (sort yards) nodes, respectively. In this network, each truckload is represented by a pair of pickup and delivery nodes. It is noteworthy that as cut blocks (sort yards) can send (receive) more than one truckload, each cut block (each sort yard) can have one or several nodes. The approach adopted in this study to model the problem has similarities with the one employed by Soares et al. (2019). In both studies, network models are used to model the scheduling problem. Unlike Soares et al. (2019), this paper synchronizes log loaders and trucks at both pickup and delivery nodes and incorporates decisions related to the trucking contractors.

For modeling purposes, the problem is divided into three sub-problems. Figure 1 represents the network for the first sub-problem at the pickup (cut block) nodes. To model the problem, a virtual depot including two nodes is added to the network. The first node ($c^{+}$) is the start node and the second node ($c^{-}$) is the sink node. Therefore, the directed graph, shown in Figure 1, can be defined using equation (1)(1) $G_1=\left(N_1,A_1\right),$(1) .

(1) $G_1=\left(N_1,A_1\right),$(1)

Figure 1. The sub-network of loaders for loading activities at cut blocks (pickup network).

In equation (1)(1) $G_1=\left(N_1,A_1\right),$(1) , $N_1=c^{+}\cup c^{-}\cup \mathit{P}$ and the nodes are connected by arcs ($A_1$). This problem can be considered as a vehicle routing problem (VRP), where each node must be exactly visited once, by only one loader. These loaders start their trip from the start node and finish their work by visiting the sink node. In the directed graph, ${\phi }_{il}^{+}$ and ${\phi }_{il}^{-}$ show the sets of nodes that succeed and precede node $\mathit{i}$ by loader $\mathit{l}$, respectively. A simple example is shown in Figure 2 where nodes $c^{-}$and $\mathit{P}3$ succeed node $\mathit{P}2$ by loader $\mathit{l}$ (i.e. ${\phi }_{P2l}^{+}\in \left\{c^{-},\mathit{P}3\right\}$), and nodes $c^{+}$, $\mathit{P}1$, and $\mathit{P}4$ precede node $\mathit{P}2$ by loader $\mathit{l}$ (i.e. ${\phi }_{P2l}^{-}\in \left\{c^{+},\mathit{P}1,\mathit{P}4\right\}$).

Figure 2. An example of the set of succeeding (${\phi }_{P2l}^{+}$) and preceding (${\phi }_{P2l}^{-}$) nodes in the directed graph.

The second sub-problem associated with the trucks is a pickup and delivery network (Figure 3). The network is presented using equation (2)(2) $G_2=\left(N_2,A_2\right),$(2) .

(2) $G_2=\left(N_2,A_2\right),$(2)

Figure 3. The sub-network of trucks traveling between cut blocks and sort yards (pickup and delivery network).

In equation (2)(2) $G_2=\left(N_2,A_2\right),$(2) , $N_2=f^{+}\cup f^{-}\cup \mathit{P}\cup \mathit{D}$ and arcs $A_2$ are the arcs that connect the nodes. Unlike the first sub-problem, the depot is a real depot in the second sub-problem. Each pickup and delivery node must be visited only once by one truck. If a truck is used, it should start the work from the start node ($f^{+}$) and return to the sink node ($f^{-}$). In the directed graph displayed in Figure 3, solid arrows show the truckloads for which the pickup and delivery nodes, responsible contractor, and the required truck type are known from the upper-level model. The dotted arrows, however, are the possible paths that each truck can choose to traverse. In other words, when a truck is at a pickup node, it has only one choice, i.e. choosing the solid arrow which means going to the destination of the truckload. However, when the truck is at the demand node, it can choose one of the succeeding dotted arrows, i.e. going to one of the pickup nodes or visiting the sink node. In this sub-network, ${\phi }_{iv}^{+}$ and ${\phi }_{iv}^{-}$ are the sets of nodes that succeed and precede node $\mathit{i}$ by truck $\mathit{v}$, respectively.

Finally, Figure 4 depicts a directed graph, defined using equation (3)(3) $G_3=\left(N_3,A_3\right)$(3) for the loaders at the delivery nodes.

(3) $G_3=\left(N_3,A_3\right)$(3)

Figure 4. The sub-network of loaders for unloading activities at sort yards (delivery network).

In equation (3)(3) $G_3=\left(N_3,A_3\right)$(3) , $N_3=g^{+}\cup g^{-}\cup \mathit{D}$ and $A_3$ are the arcs connecting different nodes. Similarly, a virtual start node ($g^{+}$) and a virtual sink node ($g^{-}$) are added to the graph and the problem is considered as a VRP, where each node should be visited only once, by a loader. Also, ${\phi }_{il}^{+}$ and ${\phi }_{il}^{-}$ are the sets of nodes succeeding and preceding node $\mathit{i}$ by loader $\mathit{l}$, respectively.

At each pickup and delivery node, the loaders and trucks have to be synchronized. It means that both machines should start a loading/unloading activity simultaneously at each node. If each of the machines arrives at a node earlier, it should wait until the other machine arrives at the node to start their work at the same time.

Mathematical model

For the scheduling problem, an MIP optimization model is developed for a one-day planning horizon to determine the routing and scheduling decisions considering synchronization of the log loaders and heterogeneous trucks at both pickup and delivery nodes. Table 4 lists notations for the proposed model. All dolar values are in Canadian dollars.

Table 4. Notations of the mathematical programming model.

Symbol	Description
Sets
$\mathrm{N}$	Set of all nodes $\mathit{i},\mathit{j}\in \left\{1,2,\dots ,\mathit{N}\right\}$
${\mathrm{N}}_1$	Subset of nodes at the pickup network $N_1\subseteq \mathit{N}$
${\mathrm{N}}_2$	Subset of nodes at the pickup and delivery network $N_2\subseteq \mathit{N}$
${\mathrm{N}}_3$	Subset of nodes at the delivery network $N_3\subseteq \mathit{N}$
$\mathrm{P}$	Set of the pickup nodes $\mathit{P}\subseteq \mathit{N}$
$\mathrm{D}$	Set of the delivery nodes $\mathit{D}\subseteq \mathit{N}$
${\mathrm{c}}^{+},\mathit{\hspace{0.17em}}{\mathrm{c}}^{-}$	The start and sink nodes for the loaders at the pickup network
${\mathrm{f}}^{+},\mathit{\hspace{0.17em}}{\mathrm{f}}^{-}$	The start and sink nodes for the trucks at the pickup and delivery network
${\mathrm{g}}^{+},\mathit{\hspace{0.17em}}{\mathrm{g}}^{-}$	The start and sink nodes for the loaders at the delivery network
${\phi }_{il}^{+}$	Set of nodes succeeding node $\mathit{i}$ by loader $\mathit{l}$
${\phi }_{il}^{-}$	Set of nodes preceding node $\mathit{i}$ by loader $\mathit{l}$
${\phi }_{iv}^{+}$	Set of nodes succeeding node $\mathit{i}$ by truck $\mathit{v}$
${\phi }_{iv}^{-}$	Set of nodes preceding node $\mathit{i}$ by truck $\mathit{v}$
$\mathrm{L}$	Set of all loaders $\mathit{l}\in \left\{1,2,\dots \mathrm{L}\right\}$
${\mathrm{L}}_P$	Subset of loaders at the pickup network $L_P\subseteq \mathit{L}$
${\mathrm{L}}_D$	Subset of loaders at the delivery network $L_D\subseteq \mathit{L}$
$\mathrm{O}$	Set of trucking contractors $\mathit{o}\in \left\{1,2,\dots ,\mathrm{O}\right\}$
$\mathrm{K}$	Set of truck types $\mathit{k}\in \left\{1,2,\dots ,\mathrm{K}\right\}$
$V_{\mathit{ok}}$	Set of trucks of type $\mathit{k}$ that contractor $\mathit{o}$ owns $\mathit{v}\in \left\{1,2,\dots ,V_{\mathit{ok}}\right\}$
Parameters
$\mathrm{F}{\mathrm{C}}_k$	Deployment cost of truck type $\mathit{k}$ ($)
$\mathrm{V}{\mathrm{C}}_k$	Unit variable cost of using truck type $\mathit{k}$ ($/minute)
$\mathrm{O}{\mathrm{C}}_k$	Unit overtime cost of using truck type $\mathit{k}$ ($/minute)
$\mathrm{DIS}\left(\mathrm{i},\mathrm{j}\right)$	Distance between node $\mathit{i}$ and node $\mathit{j}$ for loaders and dummy depots (km)
$\mathrm{D}{\mathrm{C}}_l$	Unit traveling cost of loader $\mathit{l}$ per unit of distance ($/km)
${\alpha }_{ij}^o$	If contractor $\mathrm{o}$ delivers a truckload from pickup node $\mathit{i}$ to delivery node $\mathit{j}$ the value is 1; otherwise, 0
${\beta }_{ij}^k$	If truck type $\mathrm{k}$ delivers a truckload from pickup node $\mathit{i}$ to delivery node $\mathit{j}$ the value is 1; otherwise, 0
$\mathrm{M}$	A very large number used for logical constraints
${\mathrm{D}}_{ij}^l$	Traveling time of loader $\mathit{l}$ from node $\mathit{i}$ to node $\mathit{j}$ (minute)
${\mathrm{D}}_{ij}^k$	Traveling time of truck type $\mathrm{k}$ from node $\mathit{i}$ to node $\mathit{j}$ (minute)
${\mathrm{S}}_i$	Service (loading or unloading) time at node $\mathit{i}$ (minute)
${\mathrm{T}}_{\mathit{Max}}$	Maximum daily driving time for each truck (minute)
${\mathrm{E}}_{\mathit{Max}}$	Maximum overtime that can be assigned to each truck driver (minute)
Decision variables
$x_{ij}^l$	If loader $\mathit{l}$ traverses from node $\mathit{i}$ to node $\mathit{j}$ (at the pickup network) the value is 1; otherwise, 0
$y_{ij}^{okv}$	If contractor $\mathit{o}$ uses truck $\mathit{v}$ of type $\mathit{k}$ to traverse from node $\mathit{i}$ to node $\mathit{j}$ the value is 1; otherwise, 0
$z_{ij}^l$	If loader $\mathit{l}$ traverses from node $\mathit{i}$ to node $\mathit{j}$ (at the delivery network) the value is 1; otherwise, 0
$t_i^l$	Arrival time of loader $\mathit{l}$ at node $\mathit{i}$
$t_i^{okv}$	Arrival time of truck $\mathit{v}$ of type $\mathit{k}$ owned by contractor $\mathit{o}$ at node $\mathit{i}$
$w_i$	Waiting time of trucks at node $\mathit{i}$
$u_i$	Waiting time of loaders at node $\mathit{i}$
$e^{\mathit{okv}}$	Overtime assigned to driver of truck $\mathit{v}$ of type $\mathit{k}$ owned by contractor $\mathit{o}$

Objective function

Equation 4(4) $\mathrm{Min}.\mathit{Z}=\mathit{\hspace{0.17em}}\sum _{\mathit{v}\in {\mathit{V}}_{\mathit{oK}}}\mathit{\hspace{0.17em}}\sum _{\mathit{k}\in \mathit{K}}\mathit{\hspace{0.17em}}\sum _{\mathit{o}\in \mathit{O}}\mathit{\hspace{0.17em}}\sum _{j\in {\phi }_{f^{+}\mathit{v}}^{+}}\mathrm{F}{\mathrm{C}}_k.y_{f^{+}\mathit{j}}^{\mathit{okv}}+\mathit{\hspace{0.17em}}\sum _{\mathit{v}\in {\mathit{V}}_{\mathit{ok}}}\mathit{\hspace{0.17em}}\sum _{\mathit{k}\in \mathit{K}}\mathit{\hspace{0.17em}}\sum _{\mathit{o}\in \mathit{O}}\mathrm{V}{\mathrm{C}}_k.\left(t_{f^{-}}^{\mathit{okv}}-e^{\mathit{okv}}\right)+\mathit{\hspace{0.17em}}\sum _{\mathit{v}\in {\mathit{V}}_{\mathit{oK}}}\mathit{\hspace{0.17em}}\sum _{\mathit{k}\in \mathit{K}}\mathit{\hspace{0.17em}}\sum _{\mathit{o}\in \mathit{O}}\mathrm{O}{\mathrm{C}}_k.e^{\mathit{okv}}+\mathit{\hspace{0.17em}}\sum _{l\in L_P}\mathit{\hspace{0.17em}}\sum _{\left(\mathit{i},\mathit{j}\right)\in N_1}\mathrm{D}{\mathrm{C}}_l.\mathrm{DI}{\mathrm{S}}_{\mathit{ij}}.x_{ij}^l+\mathit{\hspace{0.17em}}\sum _{l\in L_D}\mathit{\hspace{0.17em}}\sum _{\left(\mathit{i},\mathit{j}\right)\in N_3}\mathrm{D}{\mathrm{C}}_l.\mathrm{DI}{\mathrm{S}}_{\mathit{ij}}.z_{ij}^l$(4) presents the objective function of the model which is to minimize the total costs including deployment, variable, and overtime costs of using trucks, and traveling costs of loaders between nodes. It is noteworthy that incorporating the deployment cost results in a trade-off between allowing a truck driver to work overtime or dispatching a new truck.

(4) $\mathrm{Min}.\mathit{Z}=\mathit{\hspace{0.17em}}\sum _{\mathit{v}\in {\mathit{V}}_{\mathit{oK}}}\mathit{\hspace{0.17em}}\sum _{\mathit{k}\in \mathit{K}}\mathit{\hspace{0.17em}}\sum _{\mathit{o}\in \mathit{O}}\mathit{\hspace{0.17em}}\sum _{j\in {\phi }_{f^{+}\mathit{v}}^{+}}\mathrm{F}{\mathrm{C}}_k.y_{f^{+}\mathit{j}}^{\mathit{okv}}+\mathit{\hspace{0.17em}}\sum _{\mathit{v}\in {\mathit{V}}_{\mathit{ok}}}\mathit{\hspace{0.17em}}\sum _{\mathit{k}\in \mathit{K}}\mathit{\hspace{0.17em}}\sum _{\mathit{o}\in \mathit{O}}\mathrm{V}{\mathrm{C}}_k.\left(t_{f^{-}}^{\mathit{okv}}-e^{\mathit{okv}}\right)+\mathit{\hspace{0.17em}}\sum _{\mathit{v}\in {\mathit{V}}_{\mathit{oK}}}\mathit{\hspace{0.17em}}\sum _{\mathit{k}\in \mathit{K}}\mathit{\hspace{0.17em}}\sum _{\mathit{o}\in \mathit{O}}\mathrm{O}{\mathrm{C}}_k.e^{\mathit{okv}}+\mathit{\hspace{0.17em}}\sum _{l\in L_P}\mathit{\hspace{0.17em}}\sum _{\left(\mathit{i},\mathit{j}\right)\in N_1}\mathrm{D}{\mathrm{C}}_l.\mathrm{DI}{\mathrm{S}}_{\mathit{ij}}.x_{ij}^l+\mathit{\hspace{0.17em}}\sum _{l\in L_D}\mathit{\hspace{0.17em}}\sum _{\left(\mathit{i},\mathit{j}\right)\in N_3}\mathrm{D}{\mathrm{C}}_l.\mathrm{DI}{\mathrm{S}}_{\mathit{ij}}.z_{ij}^l$(4)

Constraints

Constraint sets (5) and (10) are related to the first sub-problem network for the loaders at the pickup nodes. Constraint sets (5) and (6) imply that if loaders are used, they have to return to the depot at the end of the day.

(5) $\sum _{j\in {\phi }_{c^{+}\mathit{l}}^{+}}{x}_{c^{+}\mathit{j}}^{\mathit{l}}=\sum _{i\in {\phi }_{c^{-}\mathit{l}}^{-}}{x}_{i{c}^{-}}^{\mathit{l}}\mathrm{\forall l}\in L_P$(5)

(6) $\sum _{j\in {\phi }_{c^{+}\mathit{l}}^{+}}{x}_{c^{+}\mathit{j}}^{\mathit{l}}\le 1\mathrm{\forall l}\in L_P$(6)

Constraint set (7) is the flow balancing constraint, ensuring that the loaders do not get stuck at the pickup nodes. In these constraints, if a loader enters a node, it should exit the node.

(7) $\sum _{i\in {\phi }_{jl}^{-}}{x}_{ij}^l=\sum _{i\in {\phi }_{jl}^{+}}{x}_{ji}^l\mathrm{\forall l}\in L_P,\mathit{j}\in N_1\setminus \left\{c^{+}\cup c^{-}\right\}$(7)

Arrival time of loaders at each pickup node is calculated using constraint sets (8) and (9).

(8) $S_{c^{+}}+D_{ij}^l\le t_j^l+\mathit{M}\left(1-x_{c^{+}\mathit{j}}^{\mathit{l}}\right)\mathrm{\forall l}\in L_P,\mathit{j}\in N_1\mathrm{\setminus }\left\{c^{+}\right\}$(8)

(9) $t_i^l+S_i+D_{ij}^l+u_i\le t_j^l+\mathit{M}\left(1-x_{ij}^l\right)\mathrm{\forall l}\in L_P,\mathit{i}\notin \left\{c^{+}\right\},\mathit{j}\in N_1\mathrm{\setminus }\left\{c^{+}\right\}$(9)

Constraint set (10) links the scheduling and loader assignment decisions in each pickup node.

(10) $t_j^l\le \mathit{M}\sum _{i\in {\phi }_{jl}^{-}}{x}_{ij}^l\mathrm{\forall l}\in L_P,\mathit{j}\in N_1\setminus \left\{c^{+}\cup c^{-}\right\}$(10)

Constraint sets (11)-(18) are associated with the second sub-problem network for the trucks at the pickup and delivery nodes. Constraint set (11) forces the dispatched trucks to return to their depot after delivering the assigned truckloads. Constraints (12) state that contractors have a choice to use each of their trucks or leave it at the depot.

(11) $\sum _{j\in {\phi }_{f^{+}\mathit{v}}^{+}}{y}_{f^{+}\mathit{j}}^{\mathit{okv}}=\sum _{i\in {\phi }_{f^{-}\mathit{v}}^{-}}{y}_{i{f}^{-}}^{\mathit{okv}}\mathrm{\forall o}\in \mathit{O},\mathit{k}\in \mathit{K},\mathit{v}\in V_{\mathit{ok}}$(11)

(12) $\sum _{j\in {\phi }_{f^{+}\mathit{v}}^{+}}{y}_{f^{+}\mathit{j}}^{\mathit{okv}}\le 1\mathrm{\forall o}\in \mathit{O},\mathit{k}\in \mathit{K},\mathit{v}\in V_{\mathit{ok}}$(12)

Constraint set (13) guarantees that a truck entering a node should exit the node to balance inflows and outflows.

(13) $\sum _{i\in {\phi }_{jv}^{-}}{y}_{ij}^{okv}=\sum _{i\in {\phi }_{jv}^{+}}{y}_{ji}^{okv}\mathrm{\forall o}\in \mathit{O},\mathit{k}\in \mathit{K},\mathit{v}\in V_{\mathit{ok}},\mathit{j}\in N_2\setminus \left\{f^{+}\cup f^{-}\right\}$(13)

Constraint set (14) ensures that each truckload is assigned to its contractor and compatible truck type. The information regarding the compatibility of contractors and truck types for the truckloads is obtained from the upper-level model introduced by Ghotb et al. (2022).

(14) $\sum _{\mathit{v}\in {\mathit{V}}_{\mathit{ok}}}{y}_{ij}^{okv}\le \mathit{M}.{\alpha }_{ij}^o.{\beta }_{ij}^k,\mathrm{\forall o}\in \mathit{O},\mathit{k}\in \mathit{K},\mathit{i},\mathit{j}\in N_2\setminus \left\{f^{+}\cup f^{-}\right\}$(14)

Contractors have a certain number of trucks. Constraint set (15) indicates that the total number of trucks that each contractor uses for delivering the truckloads should not exceed the number of trucks that the contractor has available.

(15) $\sum _{\mathit{v}\in {\mathit{V}}_{\mathit{ok}}}\sum _{j\in {\phi }_{f^{+}\mathit{v}}^{+}}{y}_{f^{+}\mathit{j}}^{\mathit{okv}}\le \left|V_{\mathit{ok}}\right|\mathrm{\forall o}\in \mathit{O},\mathit{k}\in \mathit{K}$(15)

Constraint sets (16) and (17) determine the arrival time of trucks at each node.

(16) $S_{f^{+}}+D_{ij}^k\le t_j^{okv}+\mathit{M}\left(1-y_{f^{+}\mathit{j}}^{\mathit{okv}}\right)\mathrm{\forall o}\in \mathit{O},\mathit{k}\in \mathit{K},\mathit{v}\in V_{\mathit{ok}},\mathit{j}\in N_2\mathrm{\setminus }\left\{f^{+}\right\}$(16)

(17) $t_i^{okv}+S_i+D_{ij}^k+w_i\le t_j^{okv}+\mathit{M}\left(1-y_{ij}^{okv}\right)\text{break}\mathrm{\forall }\mathit{o}\in \mathit{O},\mathit{k}\in \mathit{K},\mathit{v}\in V_{\mathit{ok}},\mathit{i}\notin \left\{f^{+}\right\},\mathit{j}\in N_2\mathrm{\setminus }\left\{f^{+}\right\}$(17)

The scheduling and truck assignment decisions are linked by constraint set (18).

(18) $t_j^{okv}\le \mathit{M}\sum _{i\in {\phi }_{jv}^{-}}{y}_{ij}^{okv}\mathrm{\forall o}\in \mathit{O},\mathit{k}\in \mathit{K},\mathit{v}\in V_{\mathit{ok}},\mathit{j}\in N_2\setminus \left\{f^{+}\cup f^{-}\right\}$(18)

Constraint sets (19)-(24) are related to the third sub-problem for the loaders at the delivery nodes. Constraint sets (19) and (20) state that if a loader for unloading a truck is used, it should start from the dummy depot and return to it.

(19) $\sum _{j\in {\phi }_{g^{+}\mathit{l}}^{+}}{z}_{g^{+}\mathit{j}}^{\mathit{l}}=\sum _{i\in {\phi }_{g^{-}\mathit{l}}^{-}}{z}_{i{g}^{-}}^{\mathit{l}}\mathrm{\forall l}\in L_D$(19)

(20) $\sum _{j\in {\phi }_{g^{+}\mathit{l}}^{+}}{z}_{g^{+}\mathit{j}}^{\mathit{l}}\le 1\mathrm{\forall l}\in L_D$(20)

Constraint set (21) balances inflows and outflows of each demand node for each loader.

(21) $\sum _{i\in {\phi }_{jl}^{-}}{z}_{ij}^l=\sum _{i\in {\phi }_{jl}^{+}}{z}_{ji}^l\mathrm{\forall l}\in L_D,\mathit{j}\in N_3\setminus \left\{g^{+}\cup g^{-}\right\}$(21)

The arrival times of loaders at each demand node are calculated using constraint sets (22) and (23).

(22) $S_{g^{+}}+D_{ij}^l\le t_j^l+\mathit{M}\left(1-z_{g^{+}\mathit{j}}^{\mathit{l}}\right)\mathrm{\forall l}\in L_D,\mathit{j}\in N_3\mathrm{\setminus }\left\{g^{+}\right\}$(22)

(23) $t_i^l+S_i+D_{ij}^l+\mathit{w}{l}_i\le t_j^l+\mathit{M}(1-z_{ij}^l)\mathrm{\forall l}\in L_D,\mathit{i}\notin \left\{g^{+}\right\},\mathit{j}\in N_3\mathrm{\setminus }\left\{g^{+}\right\}$(23)

In each demand node, the scheduling and loader assignment decisions are linked by constraint set (24).

(24) $t_j^l\le \mathit{M}\sum _{i\in {\phi }_{jl}^{-}}{z}_{ij}^l\mathrm{\forall l}\in L_D,\mathit{j}\in N_3\setminus \left\{g^{+}\cup g^{-}\right\}$(24)

Constraint sets (25)-(27) ensure that each node is visited by appropriate machines. Constraint sets (25) and (26) impose that each pickup and delivery node would be visited exactly once by a loader. Constraint set (27) implies that pickup and delivery nodes must be visited exactly once by one truck, meaning that all truckloads should be delivered.

(25) $\sum _{l\in L_P}\sum _{i\in {\phi }_{jl}^{-}}{x}_{ij}^l=1\mathrm{\forall j}\in \mathit{P}$(25)

(26) $\sum _{l\in L_D}\sum _{i\in {\phi }_{jl}^{-}}{z}_{ij}^l=1\mathrm{\forall j}\in \mathit{D}$(26)

(27) $\sum _{\mathit{v}\in {\mathit{V}}_{\mathit{ok}}}\sum _{\mathit{k}\in \mathit{K}}\sum _{\mathit{o}\in \mathit{O}}\sum _{i\in {\phi }_{jv}^{-}}{y}_{ij}^{okv}=1\mathrm{\forall j}\in \left(\mathit{P}\cup \mathit{D}\right)$(27)

Constraint sets (28) and (29) are synchronizing the log loaders and trucks performing a task together (loading or unloading). By these constraints, the waiting times of the loaders and trucks are calculated at pickup and delivery nodes.

(28) $\sum _{l\in L_P}{t}_i^l+u_i=\sum _{\mathit{v}\in {\mathit{V}}_{\mathit{oK}}}\sum _{\mathit{k}\in \mathit{K}}\sum _{\mathit{o}\in \mathit{O}}{t}_i^{okv}+w_i\mathrm{\forall i}\in \mathit{P}$(28)

(29) $\sum _{l\in L_D}{t}_i^l+u_i=\sum _{\mathit{v}\in {\mathit{V}}_{\mathit{oK}}}\sum _{\mathit{k}\in \mathit{K}}\sum _{\mathit{o}\in \mathit{O}}{t}_i^{okv}+w_i\mathrm{\forall i}\in \mathit{D}$(29)

Truck drivers have a maximum number of working hours, but these drivers can be assigned overtime. The amount of overtime for each truck driver is calculated using constraint set (30). Constraint set (31) states that the amount of overtime assigned to each truck driver should be less than the maximum amount of overtime that can be assigned.

(30) $t_{f^{-}}^{\mathit{okv}}\le {\mathrm{T}}_{\mathit{max}}+e^{\mathit{okv}}\mathrm{\forall o}\in \mathit{O},\mathit{k}\in \mathit{K},\mathit{v}\in V_{\mathit{oK}}$(30)

(31) $e^{\mathit{okv}}\le {\mathrm{E}}_{\mathit{max}}\mathrm{\forall o}\in \mathit{O},\mathit{k}\in \mathit{K},\mathit{v}\in V_{\mathit{oK}}$(31)

Constraint sets (32) and (33) control the domain of variables.

(32) $x_{ij}^l,y_{ij}^{okv},z_{ij}^l\in \left\{0,1\right\}\mathrm{\forall i},\mathit{j}\in \left(N_1\cup N_2\cup N_3\right),\mathit{l}\in \mathit{L},\mathit{o}\in \mathit{O},\mathit{k}\in \mathit{K},\mathit{v}\in V_{\mathit{oK}}$(32)

(33) $t_i^l,t_i^{okv},w_i,u_i,e^{\mathit{okv}}\ge 0\mathrm{\forall }\mathit{i}\in \left(N_1\cup N_2\cup N_3\right),\mathit{l}\in \mathit{L},\mathit{o}\in \mathit{O},\mathit{k}\in \mathit{K},\mathit{v}\in V_{\mathit{oK}}$(33)

Execution of the model

According to the provided information by the forest company, each truck driver has a maximum regular working time of 8.5 h (510 min). In addition, 2.5 h (150 min) of overtime can be assigned. For confidentiality, values for the cost parameters are not shown in this paper. Additionally, it was assumed that there is one stationary loader at each pickup and delivery location. Therefore, we should force the loaders to visit only those nodes that have the same locations. Hence, the distance between nodes within a location is zero, while the distance is assumed to be a very large number for nodes in different locations. This assumption avoids loaders from visiting nodes in different locations.

Similar to other papers in the literature (e.g. Flisberg et al. 2009; Melchiori et al. 2022), the proposed model is applied to test problems from our real case. Eight test problems, called TP 1 to TP 8, are solved. In the test problems, two contractors are delivering truckloads from cut blocks to three sort yards for further processing. Each contractor has three highway and three off-highway trucks. Trucks 1–6 belong to contractor 1, while contractor 2 owns trucks 7 to 12. All test problems were solved by CPLEX solver in AIMMS software package on a desktop computer with Intel® Core ^TM i7–6700 CPU, 3.40 GHz processor, and 16 GB RAM.

Results

Table 5 summarizes the test problems and results. When more truckloads need to be delivered to sort yards, transportation costs increase. In addition, using the optimization model led to improvement compared to the back-and-forth trips for all test problems because it reduced the unloaded traveling costs. However, when the size of problem increases, the solver needs more time to solve the model.

Table 5. Summary of the problem size, results, and solution time for each test problem.

Test problems	No. of truckloads	No. of nodes	No. of cut blocks	Obj. function	Improvement compared to direct delivery	CPU time (second)
TP 1	9	24	2	$10,458	10.6%	3
TP 2	10	26	2	$11,191	14%	8
TP 3	11	28	2	$12,584	14%	47
TP 4	12	30	2	$14,053	10.3%	794
TP 5	9	24	3	$13,229	4.1%	9
TP 6	10	26	3	$13,910	9.8%	11
TP 7	11	28	3	$15,253	9.7%	839
TP 8	12	30	3	$15,935	12.7%	3417

For further analysis, test problem TP 8 is investigated in detail as it is the largest sized problem. The model determined detailed truck schedules including the loading and unloading times, traveling times, waiting times at both cut blocks and sort yards, and the arrival time at the depot at the end of the day. Figure 5 shows the schedule of one of the off-highway trucks delivering two truckloads.

Figure 5. Off-highway truck schedule in TP 8.

To deliver truckloads, each contractor can assign overtime and dispatch fewer trucks. Alternatively, more trucks can be utilized to reduce overtime costs. As shown in Figure 6, both contractors assigned overtime to deliver the truckloads rather than utilizing all their trucks. This is because dispatching a new truck includes additional costs related to unloaded travel time and deployment costs. Therefore, in this problem, it would be more economical to assign overtime to existing truck drivers instead of dispatching a new truck.

Figure 6. Regular time and overtime of trucks in TP 8.

In this problem, loaders were the resources required for loading and unloading trucks at cut blocks and sort yards, respectively. As dummy depots are considered for the loaders at cut blocks (sort yards), these loaders finished their work by loading (unloading) the last truckload, and they remained at their cut block (sort yard). Hence, a detailed schedule for each loader can be determined by the model. The schedule for a loader in a cut block is shown in Figure 7 where the loader loaded four truckloads and had to wait for 177 and 167 min to load the first and the last truckloads, respectively. Similarly, Figure 8 depicts the schedule of a loader that unloaded five truckloads at sort yard 2.

Figure 7. Loader schedule at cut block 1 in TP 8.

Figure 8. Loader schedule at sort yard 2 in TP 8.

We conducted a sensitivity analysis to examine how the objective function varies when input parameters are changed. The considered parameters are the deployment cost of using trucks, variable and overtime costs of using trucks, service (loading and unloading) time, maximum driving hours, and maximum amount of overtime that can be assigned to each truck driver. As suggested by Vancas (2003), the parameters are changed by ± 20% to have a proper accuracy in the feasibility study. Figure 9 summarizes the results of the sensitivity analysis for test problem TP 8.

Figure 9. Sensitivity analysis of TP 8 when parameters are changed by 20%.

As expected, when variable and overtime costs increased (decreased) by 20%, the total costs increased (decreased) by 19%. Reducing the maximum driving time by 20% led to infeasibility as the required time for delivering a truckload to sort yard 1 would be greater than the summation of reduced maximum driving time and overtime. Also, the total costs decreased by 10% when maximum driving time increased because truckloads could be delivered during the regular driving time and the need for overtime would decrease. The effects of changing maximum allowable overtime, service time, and deployment costs were negligible. Because cut blocks and sort yards were not close to each other and it took considerable time for a truck to deliver a truckload, variable costs of trucks had a higher proportion in the total costs compared to deployment costs and changing the deployment costs could not significantly impact the total costs.

Discussion

Scheduling of log trucks without synchronizing trucks and loaders may lead to an infeasible solution because waiting times of the trucks can cause delays in delivering the truckloads that result in exceeding the maximum driving time or violating time windows. Therefore, it is important to calculate waiting times of trucks at different locations. As shown in Figure 5, the developed model in this paper allows for obtaining the detailed schedule of a truck departing and returning to the depot using continuous time representation, which leads to accurate daily plans. In some of the previous papers (e.g. El Hachemi et al. 2015; Melchiori et al. 2022), scheduling and synchronization of resources are addressed using discrete time slots. However, this may generate unnecessary waiting times. In other words, if a truck arrives after the beginning of a time slot, it should wait until the starting time of the next slot. Besides, defining discrete time slots is more challenging for this problem as the trucks are heterogeneous, and the loading time is different for each truck type. Additionally, having detailed schedules reflects the overall performance of the loaders (as shown in Figures 7 and 8).

In TP 8, for each contractor, two trucks are not used, while the other truck drivers are assigned overtime (Figure 6). Indeed, dispatching a new truck results in additional costs, which include the deployment cost of the truck and the variable cost from the depot to the cut block to pick up the truckload. In this problem, as the depot is far from cut blocks, it is more economical to pay for overtime and assign more truckloads to the existing trucks than using a new truck. Similar conclusions can be drawn from the sensitivity analysis (Figure 9) where increasing the maximum amount of overtime by 20% decreases the total transportation costs. When each truck driver can be assigned more overtime, contractors may realize cost savings from dispatching fewer trucks. It should be noted that the trucking contractors have operations in other regions. Hence, any unused trucks in one region can potentially be deployed in other regions.

Incorporating contractors in optimization models have impacts on the transportation planning. Ghotb et al. (2022) showed that considering contractors and satisfying their expectations and preferences affect decisions related to distribution of logs in a one-month planning horizon and increase the costs by 0.4%. Also, the sensitivity analysis of the current study highlights the importance of contractual details in the feasibility and profitability of the daily scheduling of trucks. The unit transportation cost, maximum driving time and overtime impact the total transportation costs.

As shown in Table 5, when the number of nodes increases, it takes more time to get the optimal solution. Thus, when trucks deliver a significant number of truckloads, the problem cannot be solved in a reasonable time using commercial software packages. As the scheduling problem needs to be solved daily, non-exact solution approaches such as metaheuristics algorithms are required for larger problems to get good solutions in a short period of time.

In this paper, we introduced an optimization model for log transportation planning at the operational level considering different complexities that can address real-world log transportation problems. Having the optimization model is useful in validation of heuristics methods for the scheduling of log trucks. According to a study by Bettinger et al. (2009), the highest level of validation of results generated by heuristics methods is to compare them with the results from mixed integer optimization models. Hence, as the proposed model is flexible and can be relaxed to consider one contractor or homogeneous trucks by having only one truck type, it can be used to validate heuristics methods for a variety of problems.

Finally, it should be noted that uncertainty and unexpected events can impact the routing and scheduling decisions in real-world situations. The breakdowns of machines can potentially result in infeasible solutions. As there is a single loader at each location, any loader breakdown makes the problem infeasible because loading the trucks cannot be done. In case of a truck breakdown, the contractor may assign the truckload to other trucks or dispatch a new truck. Furthermore, if loading/unloading takes longer, there is a chance that the problem becomes infeasible because the delays may lead to longer required overtime periods than 2.5 h for each driver. Additionally, road closure leads to infeasibility of the problem because trucks cannot fulfill their assigned truckloads. Also, if a driver is sick or unavailable on a day, it is recommended to assign the truckload to other trucks and pay overtime rather than dispatching a new truck from the depot. In case of an unexpected event, the model can be solved again by adjusting the model’s parameters according to the new information.

Conclusions

This paper presented a mixed-integer linear programming model for daily routing and scheduling of heterogeneous trucks using continuous time representation. The proposed model synchronized log loaders and trucks owned by trucking contractors at both cut blocks and sort yards by determining the detailed schedules. The performance of the model was assessed by applying it to eight test problems from the real case of a large forest company in British Columbia, Canada. The results indicated that it was more economical to pay overtime than dispatching a new truck although the unit cost of overtime was 1.5 times that of the regular time. As a result, there were some trucks that remained idle at the depot. Based on the conducted sensitivity analysis, variable costs and maximum driving time were the most sensitive parameters, while changing the deployment costs had the least impact on the total transportation costs.

When the size of the problem increases, the model will not be solvable in a reasonable time. Therefore, future research should focus on developing non-exact solution approaches to solve the problem in a short period of time. Additionally, all input parameters of the proposed model had deterministic values. However, there are uncertainty sources that may impact the log logistics planning. Hence, another avenue for future studies is to take uncertainty of input parameters into account to analyze their impacts.

Acknowledgements

This research is funded by Natural Sciences and Engineering Research Council of Canada (NSERC RGPIN-2019-04563); and Mitacs and the forest company (MITACS IT12394). We also thank the forest company for their support in obtaining data and validating our model and results.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Supplemental data

Supplemental data for this article can be accessed online at https://doi.org/10.1080/14942119.2023.2291960.

Additional information

Funding

The work was supported by the Mitacs [MITACS IT12394]; Natural Sciences and Engineering Research Council of Canada [RGPIN-2019-04563].