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Sustainability 2019, 11(11), 3127; https://doi.org/10.3390/su11113127
Article
MultiObjective Sustainable Truck Scheduling in a Rail–Road Physical Internet CrossDocking Hub Considering Energy Consumption
^{1}
LAMIH, UMR CNRS 8201, Université Polytechnique HautsdeFrance, Le Mont Houy, 59313 Valenciennes, France
^{2}
RSAID, ENSATe, University of Abdelmalek Essaadi, Tétouan 93000, Morocco
^{*}
Author to whom correspondence should be addressed.
Received: 15 May 2019 / Accepted: 29 May 2019 / Published: 3 June 2019
Abstract
:In the context of supply chain sustainability, Physical Internet (PI or $\pi $) was presented as an innovative concept to create a global sustainable logistics system. One of the main components of the Physical Internet paradigm consists in encapsulating products in modular and standardized PIcontainers able to move via PInodes (such as PIhubs) using collaborative routing protocols. This study focuses on optimizing operations occurring in a Rail–Road PIHub crossdocking terminal. The problem consists of scheduling outbound trucks at the docks and the routing of PIcontainers in the PIsorter zone of the Rail–Road PIHub crossdocking terminal. The first objective is to minimize the energy consumption of the PIconveyors used to transfer PIcontainers from the train to the outbound trucks. The second objective is to minimize the cost of using outbound trucks for different destinations. The problem is formulated as a MultiObjective MixedInteger Programming model (MOMIP) and solved with CPLEX solver using Lexicographic Goal Programming. Then, two multiobjective hybrid metaheuristics are proposed to enhance the computational time as CPLEX was time consuming, especially for large size instances: MultiObjective Variable Neighborhood Search hybridized with Simulated Annealing (MOVNSSA) and with a Tabu Search (MOVNSTS). The two metaheuristics are tested on 32 instances (27 small instances and 5 large instances). CPLEX found the optimal solutions for only 23 instances. Results show that the proposed MOVNSSA and MOVNSTS are able to find optimal and near optimal solutions within a reasonable computational time. The two metaheuristics found optimal solutions for the first objective in all the instances. For the second objective, MOVNSSA and MOVNSTS found optimal solutions for 7 instances. In order to evaluate the results for the second objective, a one way analysis of variance ANOVA was performed.
Keywords:
Physical Internet; crossdocking; Rail–Road; sustainability; truck scheduling; energy consumption; MultiObjective Programming; Lexicographic Goal Programming; hybrid metaheuristics1. Introduction
Nowadays, global optimization of the supply chain is becoming the main goal of many industrial companies, especially the logistics distribution ones. The objective is to globally reduce the economical cost and to increase the productivity while taking into consideration the social and environmental aspects. Recently, the efficiency and reactivity of the supply chains has become a big challenge for distribution companies to satisfy retailers and customer demands in terms of cost, quality and delivery time [1,2]. Therefore, with the increase of environmental constraints, supply chain sustainability has emerged as a major approach for logistics firms to enhance their economical, social and environmental sustainability [3,4,5]. Since supply chains are composed of different elements (suppliers, production plants, distribution centers, retailers and customers), a robust coordination is necessary to ensure the flexibility of the products flow. In addition to the linkage between all those components of the supply chain, the structure and the configuration of the supply chain has also a major impact on the supply chain sustainability [6,7].
Crossdocking is a type of distribution centers and it is considered as one of the most efficient distribution techniques used in supply chain management [8,9]. It consists on transferring products between inbound and outbound trucks, trains or other vehicles. The process of crossdocking starts with the unloading of the products from the inbound vehicles; then, the products may be either stored temporarily in the crossdock facility or directly transferred and loaded into the outgoing vehicles.
In order to overcome the sustainability issue and to consider all its aspects (economical, environmental and social), Physical Internet (PI or $\pi $) has been presented as a new paradigm based on a metaphor from the Digital Internet by encapsulating products in standardized modular containers called PIcontainers which are handled and moved using standard collaboration protocols inspired from TCPIP protocols of the Digital Internet [10]. The idea of the PI is to build a worldwide interconnected logistics network through PInodes (PItransits, PIhubs, etc.) enabling the external and internal routing of PIcontainers between the PInodes using PImovers (PItrucks, PIwagons, etc.) [11]. The literature of the optimization problems related to the PI concept has considerably grown over the last few years [12,13,14]. Most of the papers in the PI literature focus on the global interconnected supply chain rather than PInodes, especially crossdocking PIhubs.
In this paper, the focus is on the crossdocking terminals. There are many types of PInodes that can be identified in the PI network. For example:
 Road–Road PIhubs: Crossdocking terminals used to efficiently transfer the PIcontainers between inbound and outbound trucks through PIsorters. They are composed of PIsorters connected to the PIdocks using maneuvering areas that arrange the PIcontainers after being unloaded from inbound trucks to be routed in the PIsorters and then grouped and loaded into the outgoing trucks [15].
 Road–Rail PIhubs: Use the same mechanism as the Road–Road PIhubs. However, the PIcontainers are transferred between trucks and trains and between trains and other trains using PIsorters for routing and maneuvering areas to arrange containers. The interested reader can refer to [16] for a detailed functional design of the Road–Rail PIhubs.
 Water–Road PIhubs: Are designed to transfer the PIcontainers between boats and trucks in a port terminal [10].
We center our attention in this paper on the Rail–Road crossdocking PIhubs. The objective is to find a truck schedule that minimizes the total cost which includes the energy cost of the routing of PIcontainers through the PIconveyors and, at the same time, the cost of using outbound trucks for each different destination. The problem is formulated as a MultiObjective MixedInteger Programming model (MOMIP). The problem is then solved using two hybrid multiobjective metaheuristics: MultiObjective Variable Neighborhood Search hybridized with Simulated Annealing (MOVNSSA) and with a Tabu Search (MOVNSTS). The remainder of this manuscript is categorized as follows: In Section 2, we review the literature related to the Physical Internet, crossdock truck scheduling, sustainability, and the solving methods for multiobjective problems, especially those related to the truck scheduling in crossdocks. Section 3 presents the description and the layout of the studied Rail–Road crossdocking PIhub. The mathematical formulation of the problem is detailed in Section 4. Section 5 introduces the proposed solving methods. Numerical results are presented and analyzed in Section 6. Finally, a conclusion and possible directions for future works are presented.
2. Literature Review
This section presents the works in the literature that are related to the classical and PI crossdock truck scheduling problems with the different solving approaches used in the literature. Then, the works addressing the sustainability issue are detailed for the global supply chain and especially in crossdocking terminals. Finally, the multiobjective solving approaches used in the literature are reviewed.
2.1. Classical CrossDocking Terminals
Optimizing the crossdocking operations was widely addressed in the literature [8]. Several literature reviews and classifications were conducted concerning crossdocking optimization problems. The reader can refer to [8,9,17] for detailed classifications and reviews. Many papers addressed the crossdock scheduling with single receiving and shipping dock with different approaches. For instance, Yu and Egbelu [18] formulated the problem mathematically as a MixedInteger Programming model (MIP) to minimize the makespan. The problem was then solved using a heuristic. Later, many authors suggested different approaches to solve the single door crossdock truck scheduling problem using Genetic Algorithm [19], Hybrid Particle Swarm Optimization [20] and other various metaheuristics [21,22]. Other researches addressed the problem considering different uncertainties in various situations (taking into account the breakdown of the trucks [23], the availability of crossdock resources such as handling systems and dock doors [24], unknown trucks’ arrival time [25], etc.).
In multipledoor crossdock scheduling problem, which consists on assigning trucks to the inbound and outbound docks on a time horizon, various mathematical models were developed [8,26]. In a recent study, Gelareh et al. [27] proposed eight mathematical formulations for the crossdocking assignment problem in addition of three existing models in the literature. Then, a comparative analysis was performed on the proposed models on benchmark instances from the literature. For solving approaches, many metaheuristics were suggested: Population based metaheuristics (Differential Evolution [28,29], Diploid Differential Evolution [30], Genetic algorithm [25], Particle Swarm Optimization [31], etc.) and single solution based metaheuristics (Tabu search [32], Variable Neighborhood Search [33], Simulated Annealing [33,34], etc.).
In the multiobjective context, Golias et al. [35] proposed a formulation of the inbound truck scheduling as a biobjective and also as bilevel mixed integer programming model to minimize the total service time and the delayed completion which are conflicting objectives. Then, the two formulations were compared. A genetic algorithm and kth best algorithm were proposed because of the complexity of the problem. Authors in [23] proposed a biobjective mathematical formulation for crossdock truck scheduling to minimize the total tardiness and weighted completion time while considering trucks’ breakdown. The problem was then solved using three multiobjective metaheuristics: Nondominated Sorting Genetic Algorithm, MultiObjective Simulated Annealing and Multiobjective Differential Evolution. Mohtashami et al. [36] developed a mathematical model in which the objective is to minimize the makespan, transportation cost and the number of truck trips. They proposed two population based metaheuristics: Nondominated Sorting Genetic Algorithm (NSGAII) and the MultiObjective Particle Swarm Optimization (MOPSO). The results show that the NSGAII outperforms MOPSO. In [22], authors addressed the multicriteria crossdock scheduling through a unified objective function minimizing both earliness and tardiness of trucks. The authors proposed three population based metaheuristic: Genetic Algorithm, Particle Swarm Optimization and Differential Evolution. The results showed that Differential Evolution is the most robust algorithm since it is less sensitive to the size of the problem. Other studies considered uncertainties with multiobjective optimization and proposed biobjective bilevel approach to solve the crossdock truck scheduling with unknown truck arrival time [37].
2.2. Physical Internet CrossDocking Hubs
PI CrossDocking Hubs are a cornerstone in the Physical Internet network. Several functional designs were suggested for different PI CrossDocking Hubs. Meller et al. [15] presented a functional design for a Roadbased transit center in which the PIcontainers are transferred between inbound and outbound trucks. Another functional design was proposed by Ballot et al. [16] for Road–Rail PIhubs with various key performance indicators from the customer’s and PIhub operator’s perspective. As mentioned in the introduction, our focus is on the PIhubs, and more precisely, the Road–Rail PIhub which is considered as a PInode in the Physical Internet network. As illustrated in Figure 1 Road–Rail PIhubs are composed mainly of PIsorters, maneuvering area, and storage areas.
Many researches were conducted to study various control and optimization problems related the Road–Rail PIhubs. The PIsorters, which are used for the routing of the PIcontainers, occupies a large area of Road–Rail PIhub. Thus, several researchers addressed the PIcontainers routing problem with different approaches. Most authors focused on intelligent reactive approaches, usually exploiting multiagent systems. One of the first papers dealing with this issue belongs to Pach et al. [38] in which a multiagent model is developed for the routing process and grouping of PIcontainers into the outgoing trucks in a Rail–Road crossdocking PIhub. Different grouping strategies of the PIcontainers were proposed to minimize the evacuation time and the tardiness of trucks loading the grouped PIcontainers. One of the cornerstones of the Physical Internet paradigm is using standardized autonomous modular PIcontainers that are easy to route in the PIhubs and over the PI transportation network. In this context, an activeness concept of the PIcontainers was introduces by Sallez et al. [39] considering PIcontainer as autonomous and active products able to make decisions such as choosing the optimal routing path and to optimize the handling and moving operations especially in the crossdocking PIhubs. Hybrid control architectures were addressed in several papers, such as in [40] by evaluating the performance of the Rail–Road PIhub using simulation while considering both internal and external disruptions. An Optimized and Reactive hybrid Control Architecture (ORCA) was suggested by Vo et al. [41] through a simulation study for the PIcontainers routing in a perturbed environment using different predictive and reactive strategies. Authors in [42] proposed a simulated annealing based metaheuristic to allocate the PIcontainers to the trucks while minimizing the distance. In order to consider uncertainties, a reactive approach based on multi agent systems is suggested to handle possible perturbations in the Rail–Road PIhub. Several papers proposed mathematical formulations of the Rail–Road PIhub problems such as PIcontainers routing and truck scheduling and allocation of PIcontainers to the outgoing trucks or trains’ wagons. For instance, in [43], authors studied the truck assignment and proposed a mathematical formulation of the PIcontainers allocation and outbound trucks scheduling in the Rail–Road section of the PIhub. The objective is to minimize the distance traveled by the PIcontainers in the PIsorters to reach the outgoing trucks. Then, in order to find a solution within a reasonable time, a four steps heuristic is proposed. A mathematical model was suggested in [44] for the truck assignment in the Road–Rail section of the PIhub. The model was then solved using a Tabu Search metaheuristic.
2.3. Sustainability in Supply Chains and CrossDocks
In recent years there has been a growing interest in supply chain sustainability from all its three dimensions: economical, social and environmental. Many studies showed the importance of the supply chain practices and the coordination between its different components (supplier, production plants distribution centers and customers) on the global sustainability of the entire supply chain [6,7,45]. The sustainability issue is usually addressed from a single aspect and must integrate and combine all the three dimensions [45]. Addressing the sustainability issue by integrating all its dimensions is becoming a challenging research direction. In order to integrate the sustainability into the planning decisions of the supply chain, Mota et al. [46] proposed a multiobjective mathematical model that takes into account the three aspects of the sustainability by suggesting assessment indicators for all the three aspects. Indeed, the economical dimension is considering the total cost which includes the fixed costs, raw material costs, transportation costs and human resources. The environmental aspect is assessed through the environmental impact of different supply chain activities such as production, transportation. The social pillar is defined by a Social Benefit Indicator. The proposed model was then validated on real case study. Kong et al. [47] addressed the just in time concept in supply chain management and incorporated the environmental and economical impact into the batch scheduling problem through a mathematical model by minimizing multiple objectives: the earliness and tardiness, resources waste and environmental emission. The problem was then solved using a polynomial time algorithm. The food industry has an important effect on the sustainability of the supply chains especially from the environmental aspect. In this context, Demartini et al. [48] analyzed the sustainability in soft drink supply chains to determine the sustainable best practices and the key performance indicators that are the most related to the sustainability issues. Guo et al. [49] proposed a hybrid genetic algorithm approach to examine how the supply chain sustainability can be affected by the coordination between the production and transportation by considering several features such as departure times and transportation modes. The results showed that the coordination between the production and green transportation has an important impact on the global supply chain sustainability.
In the crossdocking context, the sustainability issue and especially energy consumption is often ignored and has not been largely addressed in the literature. To our best knowledge, few works have addressed sustainability and energy consumption in the crossdock scheduling problem. For instance, Dulebenets [30] examined the sustainable truck scheduling and presented a mathematical formulation for the scheduling of inbound and outbound trucks in a crossdock facility. Given a set of inbound and outbound trucks with their three attributed costs (handling, waiting and delayed departure costs), the objective is to find a schedule for the inbound and outbound trucks while minimizing the total cost. The mixed integer mathematical model was solved in CPLEX. Then, a Diploid Evolutionary Algorithm was proposed to solve the problem. The proposed algorithm outperforms the typical Evolutionary Algorithm from the crossdock truck scheduling literature. As another research related to energy consumption, we can refer to the work of Shahram fard and Vahdani [50] where the authors address the trucks assignment and scheduling in a crossdocking center. They proposed a biobjective mathematical model to schedule the inbound and outbound trucks taking into account the energy consumption of the forklifts. The problem is then solved using two multiobjective metaheuristics. In this paper, we try to fill the gap in the literature by addressing the sustainability issue in crossdocking terminals in the Physical Internet context.
2.4. MultiObjective Optimization Techniques
Multiobjective optimization problems are complicated to solve compared to singleobjective problems. In the crossdocking context, various objectives can be considered. Some performance indicators concern inbound and outbound trucks such as loading and unloading time, makespan and preemption cost. While other objective functions are more related to the internal handling of products such as travel distance, total product stay time and inventory level. More detailed performance indicators can be found in the literature review of Ladier and Alpan [8]. Optimizing all these objectives can be challenging especially in case of conflicting objectives, thus any improvement in a criterion will generate a loss in the other criteria. Multiobjective problems can be solved using different approaches based on aggregation techniques, for example: the weighted sum, goal programming, lexicographic method, weighted minmax and other Pareto based approaches such as Nondominated Sorting Genetic Algorithm and MultiObjective Particle Swarm Optimization [51]. As reported by Duarte et al. [52], in the context of multiobjective optimization, population based metaheuristics are widely used than single solution based metaheuristics (also called trajectory based metaheuristics). Indeed, single solution based metaheuristics such as Tabu Search (TS) [53], Greedy Randomized Adaptive Search Procedure (GRASP) [54], Simulated Annealing (SA) [55], Variable Neighborhood Search [56], Guided Local Search (GLS) [57], etc, are not used as much as population based metaheuristics (Genetic Algorithm (GA) [58], Particle Swarm Optimization (PSO) [59], Ant Colony Optimization (ACO) [60], etc.) to solve multiobjective problems [52]. For instance, Genetic Algorithms and Particle Swarm Optimization were used to solve various multiobjective optimization problems [61,62]. Moreover, authors in [63] found that Genetic and Evolutionary Algorithms are the most used, then Simulated Annealing and finally Tabu Search. However, in single solution based metaheuristics, Tabu Search (TS) and Simulated Annealing (SA) received significant attention in solving multiobjective optimization problems [64,65]. Another category of trajectory based metaheuristics is Variable Neighborhood Search which is based on the dynamic changing of the neighborhood during the search. Few researches adapted the Variable Neighborhood Search to solve multiobjective optimization problems [52]. For instance, Geiger [66] proposed a randomized VNS to solve the multiobjective flow shop scheduling by applying different neighborhood operators. Arroyo et al. [67] developed two MultiObjective Variable Neighborhood Search algorithms (MOVNS1 and MOVNS2) for the single machine scheduling problem while minimizing two conflicting objectives: earliness/tardiness and total flow time. The results of their algorithm outperformed the MOVNS proposed by Geiger [66]. Pareto based approaches are also applied in infrastructure maintenance such as pavement preservation. In this context, Lu and Tolliver [68] addressed the pavement management multiobjective optimization problem by suggesting a simulated constraint boundary method (SCBM) to optimize conflicting objectives: minimizing the total cost and maximizing the smoothness of the pavements. The proposed SCBM provides Pareto solutions without requiring any preferences parameters for the objective functions and the decision maker does not have to convert the objectives’ units to a cost or monetary unit to use the approach. The performance of the SCBM approach is then compared to the Genetic Algorithm which takes additional computational time to find the Pareto solutions. In a comparative study, Jaszkiewicz [69] addressed the biobjective set covering problem. The authors proposed a Pareto memetic algorithm (PMA) to find the set of Paretooptimal solutions. The proposed PMA was compared to ten different multiobjective metaheuristics. Yan et al. [70] proposed a biobjective fuzzy mixed integer nonlinear programming model for the hazardous materials routing problem in a Road–Rail multimodal transportation network under uncertainty and sustainability constraints. The problem was then solved using a threestage exact solution strategy.
This work aims to address the multiobjective sustainable truck scheduling problem in the Rail–Road Crossdocking PIhub. The objective is to minimize both the cost of energy consumption used by the PIconveyors for the routing of the PIcontainers as well as the cost of using outbound trucks for each different destination. A MultiObjective MixedInteger Programming model (MOMIP) is suggested for minimizing both objectives. Then, due to the complexity of the problem, two MultiObjective Variable Neighborhood Search based metaheuristics are developed and hybridized with Simulated Annealing (MOVNSSA) and Tabu Search (MOVNSTS) to make use of the features of each algorithm. The mathematical model and the two metaheuristics are then evaluated on several randomly generated instances.
3. Problem Description and Working Assumptions
This section presents a description of the Rail–Road PIhub crossdocking process with its different functionalities and layout. The Rail–Road PIhub is a type of PInodes that is used to transfer PIcontainers from the train to the outgoing trucks. It is composed of mainly a PIsorter, and two maneuvering areas in front of the train and the loading docks. The process of crossdocking starts by unloading the PIcontainers from the wagons. Then, the PIcontainer are routed to the outbound docks, grouped by destination, and then loaded into the outgoing trucks (Figure 1).
In the crossdock scheduling context, objectives and constraints that are related to sustainability were not largely addressed in the literature. To the best of our knowledge, references [30,50] are the only ones considering sustainability or energy consumption constraints in classical crossdocking terminals. Therefore, in the PI context there is still a need for research that considers sustainable objectives and constraints for the Rail–Road PIhubs crossdocking terminals. For this study, in the context of Rail–Road PIhub crossdock truck scheduling, insuring the sustainability consists on minimizing the use of the energy consumed by the PIconveyors to move PIcontainers between the outbound trucks and the train’s wagons in addition to the minimization of the cost of the trucks used to serve each destination. Therefore, the cost of using outgoing trucks must be minimized while, at the same time, minimizing the PIconveyors energy by finding the shortest path from the wagons to the outgoing trucks. The energy is then calculated by finding the number of PIconveying units swept by the PIcontainers during the way from the wagons until arriving to the trucks. Those two objectives are conflicting. Indeed, minimizing the PIconveying energy could lead to the use of multiple trucks for the same destination, which is not an optimal choice, since the cost of using additional trucks for the same destination is expensive compared to the PIconveying energy. Moreover, many trucks will leave the PIhub with empty spaces. To face this issue, MultiObjective approaches are developed and detailed in the next two sections: a MultiObjective mathematical MIP model (MOMIP) and two MultiObjective metaheuristics (MOVNSSA and MOVNSTS).
The following are the main assumptions considered for this problem:
 There are two objective functions to minimize: ${F}_{1}$ (the cost of using outbound trucks) and ${F}_{2}$ (the energy consumption cost for the PIconveyors).
 The two objective functions are arranged in some order defined by the decision maker: minimizing ${F}_{1}$ and then ${F}_{2}$ in this study for example.
 The train unloads PIcontainers that have different lengths and destinations.
 Each outbound truck must load only PIcontainers that have the same destination.
4. Mathematical Formulation
This section presents the mathematical model of the studied sustainable truck scheduling problem in a Road–Rail PIhub. The problem is formulated as MultiObjective MixedInteger Programming (MOMIP) model. The objective is to find the grouping of the PIcontainers and the assignment and scheduling of the trucks at the docks. Given the lengths, destinations and positions of the PIcontainers in the train and a set of available outbound trucks, the objective is to find the best grouping of the PIcontainers in the trucks and the trucks’ assignment and scheduling at the docks while minimizing two conflicting objectives: The energy cost for using PIconveyors to handle the PIcontainers and, at the same time, the cost of using the outgoing trucks. The main input parameters/data and output variables are presented in Figure 2.
4.1. Problem Data and Parameters
N  Total number of PIcontainers in the train 
K  Total number of the outbound docks in the Rail→Road section 
D  Total number of destinations of PIcontainers 
H  Total number of available outbound trucks 
i, j  Indices of PIcontainers to transfer from the train to the inbound trucks ($i,j=1\cdots N$) 
k  Index of the outbound docks in the Rail→Road section ($k=1\cdots K$) 
d  Index of the destinations of PIcontainers to load into the outbound trucks ($d=1\cdots D$) 
h, g  Indices of the outbound trucks ($h,g=1\cdots H$) 
C^{E}  Cost of energy consumption for one unit of PIconveyors 
${C}_{d}^{T}$  Cost of using an outbound truck for destination d 
I  Time to load one PIcontainer into an outbound truck 
V  Truck changeover time 
Y  Vertical length of the Rail→Road section 
P_{i}  Position of the bottom left corner of the PIcontainers in the wagons of the train starting from the right axis of the Rail→Road section 
R_{k}  Position of the outbound dock k starting from the right axis of the Road→Rail section 
L_{i}  Length of a PIcontainer i to unload from the train 
G_{di}  Two dimension binary matrix ($D\times N$) containing the destination of each PIcontainer, where:
$${G}_{di}=\left\{\begin{array}{cc}1\hfill & \mathrm{if}\phantom{\rule{4.pt}{0ex}}d\phantom{\rule{4.pt}{0ex}}\mathrm{is}\phantom{\rule{4.pt}{0ex}}\mathrm{the}\phantom{\rule{4.pt}{0ex}}\mathrm{destination}\phantom{\rule{4.pt}{0ex}}\mathrm{of}\phantom{\rule{4.pt}{0ex}}\mathrm{the}\phantom{\rule{4.pt}{0ex}}\mathrm{PI}\mathrm{container}\phantom{\rule{4.pt}{0ex}}i\phantom{\rule{4.pt}{0ex}}\mathrm{to}\phantom{\rule{4.pt}{0ex}}\mathrm{load}\phantom{\rule{4.pt}{0ex}}\mathrm{in}\phantom{\rule{4.pt}{0ex}}\mathrm{outbound}\phantom{\rule{4.pt}{0ex}}\mathrm{trucks}\hfill \\ 0\hfill & \mathrm{Otherwise}\hfill \end{array}\right.$$

Q  Truck capacity 
M  A sufficient big positive number. The minimum value of M must be above:
$$M>Max(2\times \mathit{Length}\phantom{\rule{4.pt}{0ex}}\mathit{of}\phantom{\rule{4.pt}{0ex}}5\phantom{\rule{4.pt}{0ex}}\mathit{wagons}\phantom{\rule{4.pt}{0ex}}\mathit{block}+Y\times Ma{x}_{i=1\dots N}\left({L}_{i}\right),\mathit{Planning}\phantom{\rule{4.pt}{0ex}}\mathit{Horizon})$$

4.2. Decision Variables
We consider the following decision variables in this MIP model:
4.2.1. Binary Variables
$${x}_{hk}=\left\{\begin{array}{cc}1\hfill & \mathrm{if}\phantom{\rule{4.pt}{0ex}}\mathrm{the}\phantom{\rule{4.pt}{0ex}}\mathrm{outbound}\phantom{\rule{4.pt}{0ex}}\mathrm{truck}\phantom{\rule{4.pt}{0ex}}h\phantom{\rule{4.pt}{0ex}}\mathrm{is}\phantom{\rule{4.pt}{0ex}}\mathrm{assigned}\phantom{\rule{4.pt}{0ex}}\mathrm{to}\phantom{\rule{4.pt}{0ex}}\mathrm{the}\phantom{\rule{4.pt}{0ex}}\mathrm{outbound}\phantom{\rule{4.pt}{0ex}}\mathrm{dock}\phantom{\rule{4.pt}{0ex}}k\hfill \\ 0\hfill & \mathrm{Otherwise}\hfill \end{array}\right.$$
$${p}_{ih}=\left\{\begin{array}{cc}1\hfill & \mathrm{if}\phantom{\rule{4.pt}{0ex}}\mathrm{PI}\mathrm{container}\phantom{\rule{4.pt}{0ex}}i\phantom{\rule{4.pt}{0ex}}\mathrm{is}\phantom{\rule{4.pt}{0ex}}\mathrm{assigned}\phantom{\rule{4.pt}{0ex}}\mathrm{to}\phantom{\rule{4.pt}{0ex}}\mathrm{the}\phantom{\rule{4.pt}{0ex}}\mathrm{outbound}\phantom{\rule{4.pt}{0ex}}\mathrm{truck}\phantom{\rule{4.pt}{0ex}}h\hfill \\ 0\hfill & \mathrm{Otherwise}\hfill \end{array}\right.$$
$${a}_{hd}=\left\{\begin{array}{cc}1\hfill & \mathrm{if}\phantom{\rule{4.pt}{0ex}}d\phantom{\rule{4.pt}{0ex}}\mathrm{is}\phantom{\rule{4.pt}{0ex}}\mathrm{the}\phantom{\rule{4.pt}{0ex}}\mathrm{destination}\phantom{\rule{4.pt}{0ex}}\mathrm{of}\phantom{\rule{4.pt}{0ex}}\mathrm{the}\phantom{\rule{4.pt}{0ex}}\mathrm{outbound}\phantom{\rule{4.pt}{0ex}}\mathrm{truck}\phantom{\rule{4.pt}{0ex}}h\hfill \\ 0\hfill & \mathrm{Otherwise}\hfill \end{array}\right.$$
$${v}_{h}=\left\{\begin{array}{cc}1\hfill & \mathrm{if}\phantom{\rule{4.pt}{0ex}}\mathrm{outbound}\phantom{\rule{4.pt}{0ex}}\mathrm{truck}\phantom{\rule{4.pt}{0ex}}h\phantom{\rule{4.pt}{0ex}}\mathrm{is}\phantom{\rule{4.pt}{0ex}}\mathrm{used}\hfill \\ 0\hfill & \mathrm{Otherwise}\hfill \end{array}\right.$$
$${n}_{hg}=\left\{\begin{array}{cc}1\hfill & \mathrm{if}\phantom{\rule{4.pt}{0ex}}\mathrm{outbound}\phantom{\rule{4.pt}{0ex}}\mathrm{trucks}\phantom{\rule{4.pt}{0ex}}h\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}g\phantom{\rule{4.pt}{0ex}}\mathrm{are}\phantom{\rule{4.pt}{0ex}}\mathrm{asigned}\phantom{\rule{4.pt}{0ex}}\mathrm{to}\phantom{\rule{4.pt}{0ex}}\mathrm{the}\phantom{\rule{4.pt}{0ex}}\mathrm{same}\phantom{\rule{4.pt}{0ex}}\mathrm{outbound}\phantom{\rule{4.pt}{0ex}}\mathrm{dock}\hfill \\ & \phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}h\phantom{\rule{4.pt}{0ex}}\mathrm{is}\phantom{\rule{4.pt}{0ex}}\mathrm{a}\phantom{\rule{4.pt}{0ex}}\mathrm{predecessor}\phantom{\rule{4.pt}{0ex}}\mathrm{of}\phantom{\rule{4.pt}{0ex}}g\hfill \\ 0\hfill & \mathrm{Otherwise}\hfill \end{array}\right.$$
4.2.2. Integer Variables
z_{ih}  The area (Number of PIconveyor units) swept by PIcontainer i to arrive at its destination at the outbound truck h 
4.2.3. Continuous Variables
b_{h}  Start time for loading outbound truck h 
q_{h}  End time for loading outbound truck h (completion time) 
4.3. Objective Function
The objective function of the this MOMIP model is to minimize two conflicting objectives:
where:
$$\begin{array}{ccc}\mathrm{Minimize}:\hfill & \hfill \phantom{\rule{2.em}{0ex}}{F}_{1}& =\sum _{h=1}^{H}\sum _{d=1}^{D}{C}_{d}^{T}{a}_{hd}\hfill \\ \mathrm{Minimize}:\hfill & \hfill \phantom{\rule{2.em}{0ex}}{F}_{2}& ={C}^{E}\sum _{i=1}^{N}\sum _{h=1}^{H}{z}_{ih}\hfill \end{array}$$
F_{1}  The total cost of trucks used for all destination 
F_{2}  The total energy consumption cost for PIconveyors 
4.4. Constraints
$$\sum _{h=1}^{H}{p}_{ih}=1\phantom{\rule{2.em}{0ex}}(\forall i=1\dots N)$$
$$\sum _{i=1}^{N}{p}_{ih}{L}_{i}\le Q\phantom{\rule{2.em}{0ex}}(\forall h=1\dots H)$$
$${p}_{ih}+{p}_{jh}\le \sum _{d=1}^{D}{G}_{di}{G}_{dj}+1\phantom{\rule{2.em}{0ex}}(\forall i,j=1\dots N,\forall h=1\dots H:i\ne j)$$
$${p}_{ih}\le {v}_{h}\phantom{\rule{2.em}{0ex}}(\forall i=1\dots N,\forall h=1\dots H)$$
$${a}_{hd}\le {G}_{di}+1{p}_{ih}\phantom{\rule{2.em}{0ex}}(\forall i=1\dots N,\forall h=1\dots H,\forall d=1\dots D)$$
$${v}_{h}=\sum _{d=1}^{D}{a}_{hd}\phantom{\rule{2.em}{0ex}}(\forall h=1\dots H)$$
$${z}_{ih}\ge 2{P}_{i}{R}_{k}+Y{L}_{i}M(2({p}_{ih}+{x}_{hk}))\phantom{\rule{2.em}{0ex}}(\forall i=1\dots N,\forall k=1\dots K,\forall h=1\dots H)$$
$$\sum _{k=1}^{K}{x}_{hk}={v}_{h}\phantom{\rule{2.em}{0ex}}(\forall h=1\dots H)$$
$${x}_{hk}+{x}_{gk}1\le {n}_{hg}+{n}_{gh}\phantom{\rule{2.em}{0ex}}(\forall k=1\dots K,\forall h,g=1\dots H,h\ne g)$$
$${n}_{hg}+{n}_{gh}\le 1\phantom{\rule{2.em}{0ex}}(\forall h,g=1\dots H)$$
$${b}_{g}=Max(0,{q}_{h}+VM(1{n}_{hg}))\phantom{\rule{2.em}{0ex}}(\forall h,g=1\dots H)$$
$${q}_{h}={b}_{h}+I\sum _{i=1}^{N}{p}_{ih}\phantom{\rule{2.em}{0ex}}(\forall h=1\dots H)$$
$${x}_{hk},{p}_{ih},{a}_{hd},{v}_{h},{n}_{h,g}\in \{0,1\}\phantom{\rule{2.em}{0ex}}(\forall i=1\dots N,\forall k=1\dots K,\forall d=1\dots D,\forall h,g=1\dots H)$$
$${z}_{ih}\ge 0,{b}_{h}\ge 0,{q}_{h}\ge 0\phantom{\rule{2.em}{0ex}}(\forall i=1\dots N,\forall h=1\dots H)$$
4.5. Constraints Description
Constraint (2) checks that each PIcontainer is assigned to only one outbound truck. Constraint (3) ensures that the outbound trucks’ capacity is not exceeded by the PIcontainers. Constraint (4) ensures that each outbound truck, if used, can load only PIcontainers with the same destination. Constraint (5) guarantees that an outbound truck is used if at least one PIcontainer is assigned to it. Constraint (6) finds the destination of a used outbound truck based on the PIcontainers that are assigned to it. Constraint (7) is to guarantee that each used truck has only one destination. Constraint (8) computes the number of PIconveyors units swept by a PIcontainer to arrive at its destination in the outbound trucks. Constraint (9) ensures that each inbound truck has to be assigned to only one dock. Constraints (10) and (11) handle the correct relationship between the assignment variable (${x}_{hk}$) and the sequencing variable (${p}_{ih}$). Constraint (12) calculates the staring time of loading outbound trucks taking into account the end time of loading previous trucks if they are processed at the same dock. Constraint (13) calculates the end time of unloading the outbound trucks. Constraint (14) guarantees that the decision variables: ${x}_{hk},{p}_{ih},{a}_{hd},{v}_{h},{n}_{hg}$ are binary. Finally, Constraint (15) ensures that the decision variables: ${z}_{ih},{b}_{h},{q}_{h}$ are positive.
5. Solving Methods
Due to high computational times, solving the mathematical model is not practical for large size instances. Therefore, developing metaheuristics algorithms can provide near optimal or optimal solutions within reasonable computational times. Metaheuristics are widely used in the literature to solve multiobjective optimization problems [71,72,73,74], using different techniques such as the weighted sum, exponential weighted criterion, lexicographic method, weighted minmax method, etc. The solving methods proposed in this paper are based on the Lexicographic Goal Programming method, which consists on ordering the objective functions in order of importance [51]. In Lexicographic Goal Programming, the first objective is considered the most important criterion, and it is worth any decreasing in the other objectives to improve the first criterion. The second objective is considered the most important one after the first criterion, and so on. The last objective is the least important among all the previous criteria.
In this paper, two hybrid metaheuristics are proposed: MultiObjective Variable Neighborhood Search hybridized with Simulated Annealing (MOVNSSA) and the second one uses VNS hybridized with Tabu Search (MOVNSTS). The VNS is chosen due its ability to dynamically change the neighborhood search. The hybridization with Tabu Search gives the VNS the ability to avoid the local search moves already performed. While Simulated Annealing lets the VNS to accept solutions at the beginning of the search even with high deviations and then the probability of accepting solutions is decreased and the algorithm become very selective in solutions. First, a construction heuristic (${H}_{0}$) is proposed to build an initial solution by minimizing: on the one hand, the cost of used trucks by grouping the PIcontainers in the trucks based on the Best Fit Decreasing algorithm while taking into consideration the destination constraints; on the other hand, finding an initial position of the trucks at the docks using the positions of the PIcontainers to unload from the train. The main idea is to find a grouping of the PIcontainers in the outgoing trucks to optimize the first objective ${F}_{1}$, then, the generated solution is considered as an input or starting point for the two metaheuristics (MOVNSSA and MOVNSTS) that try to optimize the second objective ${F}_{2}$. In the following, a detailed description of the steps of each algorithm is presented (Construction Heuristic ${H}_{0}$, MOVNSSA and MOVNSTS).
5.1. Construction Heuristic
Since the proposed metaheuristics are based on the VNS which is a single solution based metaheuristic, it is necessary to build an initial solution as a starting point for the proposed hybrid metaheuristics. Therefore, a construction heuristic (Algorithm 1) is proposed to find an initial grouping of PIcontainers and then the assignment and scheduling of the outgoing trucks at the docks. The first step of the heuristic is to group the PIcontainers in the trucks after sorting them in a decreasing order depending on their lengths while taking into consideration the destination of each PIcontainer. In the second step, the average position of each truck is calculated depending on the position of the PIcontainers that are assigned to that truck. This average position will be used later to find the dock on which the truck will be assigned for loading the PIcontainers. The next step is to group and assign the PIcontainers to the trucks by destination. If a truck does not have enough space or has a different destination, a new truck must be selected. The last step is to calculate the schedule of the trucks (starting/ending time of processing).
Algorithm 1 Overview of the heuristic (${H}_{0}$) algorithm for the initial solution 

5.2. Neighborhood Operators
In both metaheuristics (MOVNSSA and MOVNSTS), there are three neighborhood operators performed at each inner iteration: Insertion, Swap and Insertion→Swap (Figure 3). The insertion consists on selecting a random truck in the current neighborhood and then assigning the selected truck to another different random dock. The swap operator selects two different trucks and swaps their assignments. Finally, in the last operator (Insertion→Swap), a random truck is selected, inserted in a different dock, and then swapped with a random truck. All those three neighborhood operators are performed with equal probabilities at each iteration.
5.3. MOVNSSA
The overall framework of the MOVNSSA algorithm starts by loading the initial solution of the construction heuristic (Algorithm 2). Then a set of Nb_N neighborhood structures is generated. For each selected structure, the algorithm runs the Local Search based on the Simulated Annealing (SA) algorithm for Max_VNS iterations. At each iteration the temperature parameter T is set to its initial value T_Max. During the search, the temperature T is decreased by $\mathsf{\Delta}t$ after performing each Local Search move (Insertion, Swap and Insertion→Swap) with equal probabilities for each move. The new generated solutions (${S}^{\prime}$) are accepted as current solution with a probability p:
$$p=exp\left(\frac{S{S}^{\prime}}{T}\right)$$
At the beginning, when T is still higher, the algorithm tends to accepts deteriorating moves. Then, as the temperature T is decreased, the algorithm becomes very selective on the new generated solutions (${S}^{\prime}$). This process is repeated for Max_VNS iterations and for each neighborhood structure. A shaking of the PIcontainers assignments between the trucks with the same destination is performed after Max_Shake iterations. At each iteration of the Simulated Annealing, a random solution is generated in the current neighborhood structure, the generated solution (${S}^{\prime}$) is accepted with the probability p as the current solution (S). Then, if ${S}^{\prime}<{S}_{BEST}$, ${S}^{\prime}$ is considered as the current best solution found. The inner SA loop, which is repeated for Max_VNS iterations, stops once $T\le \u03f5$. This process is repeated for each neighborhood structure. The algorithm stops after exploring all the neighborhood structures.
Algorithm 2 Overview of the hybrid MOVNSSA algorithm 

5.4. MOVNSTS
In the MOVNSTS metaheuristic, the VNS is hybridized with Tabu Search which, contrary to the Simulated Annealing, makes use of memory. Indeed, at each iteration of the Tabu Search, each performed move is stored in a Tabu List so at the next iterations those moves will not be performed for a certain number of iterations. This mechanism helps the algorithm to learn from the past moves to avoid the deteriorating ones. An aspiration criterion is defined as the deviation between the new found solution (${S}^{\prime}$) and the current solution (S) to prevent the algorithm from local optima. As described in Algorithm 3, the searching process starts by loading the initial solution from the construction heuristic. The PIcontainers assignment is shaked between the trucks having the same destination after Max_Shake iterations. Then, the Tabu Search algorithm is performed for Max_TS iterations for each neighborhood structure. At each iteration, a new random solution is generated in the selected neighborhood (${S}^{\prime}$). If the aspiration criterion is checked for the new solution, the later is selected as a current solution ($S={S}^{\prime}$). If ${S}^{\prime}<{S}_{BEST}$, then the new solution is selected as the current best solution. The algorithm ends after repeating this process for all the neighborhood structures. The main steps of the MOVNSTS algorithm are presented in Algorithm 3.
Algorithm 3 Overview of the hybrid MOVNSTS algorithm 

6. Computational Experiments
After a brief description of the implementation of the mathematical model and the development of the two metaheuristics, this section presents the obtained results on the randomly generated instances with an analysis of the obtained results.
6.1. Implementation and Instances
This section aims to evaluate the performance of the proposed metaheuristics on several randomly generated instances. First, the multiobjective mathematical model is implemented and solved using Lexicographic Goal Programming optimization in IBM CPLEX solver (Version 12.9). A time limit of one hour (3600s) is set for the solver. The heuristic (${H}_{0}$) and the two metaheuristics (MOVNSSA and MOVNSTS) are developed in C++ and all the experiments are performed on an Intel${}^{\left(R\right)}$ Core${}^{\left(TM\right)}$ i3 processor with 4 GB of RAM. The tests are performed in 5 replications for each instance. The average value is presented. The proposed algorithms are tested on a set of small and large instances randomly generated by varying several parameters. The values of the parameters are summarized in Table 1. The tuning of the metaheuristics parameters is presented in Table 2.
6.2. Numerical Results
Table 3 shows the values of both objective functions: ${F}_{1}$ (Cost of used trucks per destination) and ${F}_{2}$ (Energy cost consumption). The first four columns present the parameters of the generated instances. The next three columns show the results obtained after solving the mathematical model using Lexicographic Goal Programming in CPLEX (${F}_{1}$ and then ${F}_{2}$). The computational times are presented in the seventh column. CPLEX was able to determine the optimal solution for 23 instances. In the last four instances (marked with “*” in Table 3), CPLEX exceeds the time limit of 3600s without providing any feasible solution. The remaining of the columns shows the results of the two metaheuristics (MOVNSSA and MOVNSTS) which provide optimal results for the first objective ${F}_{1}$ and near optimal values for the second objective ${F}_{2}$ within fast computational times compared to the ones of CPLEX. The optimal values are presented in bold. In order to evaluate more the performance of the two metaheuristics, several large instances are generated. Parameters are modified for those large sized instances (Max_VNS = 1500, $\mathsf{\Delta}t$ = 0.002 and Max_TS = 500). The obtained results are presented in Table 4, and illustrated in Figure 4.
As it can be seen in Figure 5, the PIconveyors’ energy consumption, which represented by objective function ${F}_{2}$, increases significantly with the number of PIcontainers.
6.3. Sensitivity Analysis
6.3.1. ANOVA Measures
The two metaheuristics MOVNSSA and MOVNSTS found optimal values for the first objective function ${F}_{1}$. In order to show the insignificant difference for the second objective ${F}_{2}$, a testing of hypothesis using analysis of variance ANOVA is performed for both metaheuristics. The ANOVA test showed that both algorithms do not differ from the optimal value of ${F}_{2}$ at 95% of confidence limit. The detailed results are presented in Table 5. As pvalue $>0.05$, there is not a significant difference between the metaheuristics and the optimal values of CPLEX.
6.3.2. Convergence Behavior
In order to illustrate the convergence behavior of the two metaheuristics, multiple tests are performed with different values of the maximum number of iterations for each neighborhood structure (Max_VNS) which goes from 1 to 500 iterations. The obtained results are presented in Figure 6 which shows the impact of the value of the number of iterations (Max_VNS) on the total average deviation for the second objective (${F}_{2}$).
6.4. Discussion
The results obtained by both metaheuristics (MOVNSSA and MOVNSTS) seems to be promising in terms of quality of the obtained solutions especially for the first objective ${F}_{1}$ for which the optimal value was found for all the instances and near optimal and several optimal values for the second objective ${F}_{2}$. The results seems to be positive also in terms of computational time even for large sized instances.
Within the context of multiobjective optimization, using the Lexicographic Programming method for the MOMIP model and the two metaheuristics (MOVNSSA and MOVNSTS) requires a priori knowledge about the objectives which is based on the decision makers’ preferences. Therefore, the two objective functions presented in Table 3 and Table 4 are solved in a lexicographic arranged order (which is ${F}_{1}$ and then ${F}_{2}$ in this study).
Taking those points into account, the proposed methods provide only solutions that are based on the decision maker’s preferences and cannot provide a set of Pareto optimal solutions to be proposed to the decision maker. Moreover, the proposed metaheuristics are not applicable to find an alternative solution in case of any changes in the relative importance of the objectives.
7. Conclusions
This paper addressed the multiobjective sustainable truck scheduling in the Road–Rail PIhub crossdock. The problem was formulated as a MultiObjective Mixed Integer Programming model (MOMIP) considering two different objectives. The first objective minimizes the energy consumption cost for the routing of the PIcontainers using PIconveyors. The second one minimizes the cost of using the outgoing trucks for each destination. The model was then solved using Lexicographic Goal Programming in CPLEX solver. Due to the long computational times, two MultiObjective hybrid metaheuristics were proposed: MOVNSSA and MOVNSTS. The MOMIP and the two metaheuristics were evaluated on 27 small instances and 5 additional large instances. CPLEX found optimal solutions for only 23 instances. The obtained results showed that the two metaheuristics were able to generate near optimal and optimal solutions within short computational times. The results were validated through an ANOVA analysis.
In this study we used a lexicographic based method, which requires a priori knowledge from the decision maker. As an important direction of this work, we intend to extend our study by developing other Pareto based approaches that do not require a priori preferences about the relative importance of the objective functions. Those Pareto based approaches can be developed using population based metaheuristics such as NSGA, MOPSO and MOGA [75]. As another possible direction of this work, integrating external simulators could be interesting for taking into account the possible perturbations that can occur in the PIhub facility such as trucks delays, customers changing orders at the last minute, etc.
Author Contributions
Conceptualization, T.C., A.B., M.R. and D.T.; data curation, T.C.; formal analysis, T.C.; funding acquisition, A.B., M.R. and D.T.; investigation, T.C.; methodology, T.C., A.B., M.R. and D.T.; project administration, A.B., M.R. and D.T.; software, T.C.; supervision, A.B., M.R. and D.T.; validation, T.C.; writing–original draft, T.C.; writing–review and editing, T.C., A.B., M.R. and D.T.
Acknowledgments
This research was supported by the ELSAT2020 project of CPER sponsored by the French ministry of sciences, the Hauts de France region and the FEDER. This work was supported also by the ANR PINUTS Project (grant ANR14CE270015).
Conflicts of Interest
The authors declare no conflict of interest.
Abbreviations
The following abbreviations are used in this manuscript:
ACO  Ant Colony Optimization 
GA  Genetic Algorithm 
GLS  Guided Local Search 
GRASP  Greedy Randomized Adaptive Search Procedure 
LS  Local Search 
MOGA  MultiObjective Genetic Algorithm 
MOMIP  MultiObjective Mixed Integer Programming Model 
MOPSO  MultiObjective Particle Swarm Optimization 
MOVNSSA  MultiObjective Variable Neighborhood Search  Simulated Annealing 
MOVNSTS  MultiObjective Variable Neighborhood Search  Tabu Search 
NSGA  Nondominated Sorting Genetic Algorithm 
PI  Physical Internet 
PMA  Pareto Memetic Algorithm 
PSO  Particle Swarm Optimization 
SA  Simulated Annealing 
SCBM  Simulated Constraint Boundary Method 
TL  Tabu List 
TS  Tabu Search 
VNS  Variable Neighborhood Search 
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Figure 2.
Input parameters and output variables of the MultiObjective MixedInteger Programming (MOMIP) model.
Figure 3.
Neighborhood operators in MultiObjective Variable Neighborhood Search hybridized with Simulated Annealing (MOVNSSA) and MultiObjective Variable Neighborhood Search hybridized with Tabu Search (MOVNSTS): (a) Insertion (b) Swap (c) Insertion→Swap.
Figure 5.
Objective functions in CPLEX, MOVNSSA and MOVNSTS for both objectives ${F}_{1}$ and ${F}_{2}$.
Data  Abbreviations  Values 

Number of PIcontainers  N  [4, 12] 
Number of destinations  D  [1, 3] 
Number of docks  K  15 
Lengths of PIcontainers  ${L}_{i}$  {1, 2, 3, 4, 5, 10} 
Cost of energy consumption for one PIconveyor  ${C}^{E}$  0.5 
Cost of using a truck for each destination  ${C}_{d}^{T}$  [200, 800] 
Positions of PIcontainers in the train  ${P}_{i}$  [1, 75] 
Destination of PIcontainers in the train  ${G}_{di}$  Random (Binary) 
Algorithm Parameters  Abbreviations  Values 

Number of iterations for each neighborhood  Max_VNS  500 
Number of generated neighborhoods  Nb_N  3 
Number of iterations before shaking PIcontainers’ assignments in trucks  Max_Shake  10 
Simulated Annealing temperature decreasing step  $\mathsf{\Delta}t$  0.01 
Initial temperature in the Simulated Annealing  T_Max  1 
Tabu List Size  TL_Size  10 
Number of iterations for the Tabu Search process  Max_TS  100 
Instances  CPLEX  MOVNSSA  MOVNSTS  

#  D  N  H  F${}_{1}$  F${}_{2}$  Time (s)  F${}_{1}$  F${}_{2}$  Time (s)  F${}_{1}$  F${}_{2}$  Time (s)  
1  1  4  4  351  136  0.11  351  136  0.078  351  136  0.081  
2  1  5  4  750  199  0.42  750  199  0.087  750  199  0.091  
3  1  6  5  1166  152  0.69  1166  192  0.088  1166  192  0.081  
4  1  7  5  1263  218  11.36  1263  282  0.084  1263  287  0.094  
5  1  8  5  1821  234  2.78  1821  273  0.094  1821  273  0.100  
6  1  9  7  891  310  14.86  891  357  0.116  891  360.2  0.144  
7  1  10  7  1588  299  10.86  1588  431  0.134  1588  436  0.138  
8  1  11  7  1276  310  13.36  1276  406  0.122  1276  406  0.138  
9  1  12  7  2035  377  38.28  2035  494  0.140  2035  495  0.143  
10  2  4  4  1096  136  0.14  1096  136  0.056  1096  136  0.078  
11  2  5  4  1835  189  0.28  1835  194  0.066  1835  194  0.078  
12  2  6  5  1033  183  0.30  1033  183  0.081  1033  183  0.091  
13  2  7  5  1792  252  0.52  1792  306  0.084  1792  306  0.091  
14  2  8  5  2510  243  1.69  2510  283  0.091  2510  283  0.112  
15  2  9  7  1650  324  453.09  1650  355.8  0.112  1650  366  0.125  
16  2  10  7  2098  362  277.84  2098  392  0.194  2098  394.2  0.131  
17  2  11  7  1876  367  40.52  1876  400  0.147  1876  406.2  0.138  
18  2  12  7  2688  371  186.89  2688  499  0.159  2688  499  0.153  
19  3  4  4  1542  108  0.16  1542  111  0.059  1542  111  0.078  
20  3  5  4  1923  206  2.09  1923  214  0.069  1923  214  0.094  
21  3  6  5  1440  160  0.25  1440  160  0.084  1440  160  0.094  
22  3  7  5  1797  293  17.17  1797  293  0.084  1797  293  0.094  
23  3  8  5  2520  281  164.42  2520  281  0.091  2520  281  0.106  
24  3  9  7  *  *  *  2331  372.8  0.106  2331  374.8  0.128  
25  3  10  7  *  *  *  3102  422.6  0.116  3102  425.6  0.144  
26  3  11  7  *  *  *  2042  401.4  0.122  2042  407  0.141  
27  3  12  7  *  *  *  3699  495  0.141  3699  498.6  0.150 
Instances  MOVNSSA  MOVNSTS  

#  D  N  H  F${}_{1}$  F${}_{2}$  Time (s)  F${}_{1}$  F${}_{2}$  Time (s)  
28  7  20  15  5001  772.6  4.322  5001  750  4.503  
29  10  20  15  7338  801.2  4.5872  7338  776  4.7218  
30  7  30  20  5081  1279.8  7.4594  5081  1216  7.3938  
31  10  30  20  6401  1284.8  7.35  6401  1201  7.4782  
32  15  30  20  9529  1220.4  8.1656  9529  1182  8.2032 
MetaHeuristics  F  pValue 

MOVNSSA  1.623  0.209 
MOVNSTA  0.726  0.196 
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