sustainability Article
A Hybrid Estimation of Distribution Algorithm for Multi-Objective Multi-Sourcing Intermodal Transportation Network Design Problem Considering Carbon Emissions Shou-feng Ji and Rong-juan Luo * School of Business Administration, Northeastern University, Shenyang 110167, China;
[email protected] * Correspondence:
[email protected]; Tel.: +86-024-8368-0613 Received: 18 April 2017; Accepted: 26 June 2017; Published: 30 June 2017
Abstract: The increasing concern on global warming is prompting transportation sector to take into account more sustainable operation strategies. Among them, intermodal transportation (IT) has already been regarded as one of the most effective measures on carbon reductions. This paper focuses on the model and algorithm for a certain kind of IT, namely multi-objective multi-sourcing intermodal transportation network design problem (MO_MITNDP), in which carbon emission factors are specially considered. The MO_MITNDP is concerned with determining optimal transportation routes and modes for a series of freight provided by multiple sourcing places to find good balance between the total costs and time efficiencies. First, we establish a multi-objective integer programming model to formulate the MO_MITNDP with total cost (TTC) and maximum flow time (MFT) criteria. Specifically, carbon emission costs distinguished by the different transportation mode and route are included in the cost function. Second, to solve the MO_MITNDP, a hybrid estimation of distribution algorithm (HEDA) combined with a heterogeneous marginal distribution and a multi-objective local search is proposed, in which the from the Pareto dominance scenario. Finally, based on randomly generated data and a real-life case study of Jilin Petrochemical Company (JPC), China, simulation experiments and comparisons are carried out to demonstrate the effectiveness and application value of the proposed HEDA. Keywords: multi-objective optimization; intermodal transportation network design; hybrid estimation of distribution algorithm; carbon reduction; Pareto dominance
1. Introduction As one of the most important logistics activities, freight transportation which happens throughout the whole production circulation process from material purchase to finished product sale is crucial to the economic growth and market trade. With the rapid growth of global purchasing, transportation has played an increasingly important role in world resource integrations. On the other hand, due to the downsides such as energy consuming and air pollution, transportation is always located at the crossroads of economic and environmental concerns. For example, as pointed by Environmental Protection Agency (USA), transportation is the second largest contributor of carbon emission, causing about 28% of the total emissions in US in the past years [1]. Accumulation of greenhouse gas in the atmosphere, especially CO2 , has been a main source of global warming and attracts increasing concerns by different sectors [2]. For example, the European Commission announced that the major objective corresponds to transportation sector is to reduce the carbon emissions at least 60% of 1990 levels by 2050 [3]. In the past years, carbon emissions from other transportation modes have experienced a relatively slight decreasing while the road transportation is keeping a rapid growth. Road
Sustainability 2017, 9, 1133; doi:10.3390/su9071133
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transportation is still the dominant mode in inland freight transportation system, and meanwhile it is also a carbon intensive mode which contributes two-thirds of total carbon emissions in transportation sector [4]. With the growth of transportation volume and economic, Zheng and Wang 2010 has forecasted that carbon emissions from the road freight transportation will keep a continual increase by 2020 globally [5]. In China, the current energy structure is dominated by coal (about 64%) and oil (about 18%) [2], which also has been seen as the hardest obstacle to the low-carbon transportation. In view of these, different measures and regulations have been established, such as developing new energy technologies, prompting green transportation equipment, issuing relevant laws, etc. According to Benjaafar et al., 2013 [6], mode selection and route design are of great importance for low-carbon transportation systems, which might be more significant than policies, regulations, laws established by national governments at different levels or international organizations. In this regard, intermodal transportation (IT) has been put on the agenda by many countries and regions, as well as been promoted as a highly desirable measure with respect to improving the resource, environment and ecology sustainability. Unlike other conventional transportation patterns, in which the transportation modes are independently operated without any coordination, IT aims at integrating more than one mode together such that it can provide more flexible “door to door” service according to the personalized requirements of customers [7]. The optimization of IT is essentially to provide a satisfied solution for the intermodal transportation company with reducing transportation cost, transportation time and carbon emissions simultaneously. For carbon emissions reduction, it might be explained that the heavier emission modes are replaced by the lighter ones [8,9]. In general, there are three kinds of modes in IT, i.e., waterway, railway and road, and the former two usually contribute less carbon emissions than the latter. However, intermodal transportation systems have an intractable drawback that it is more likely to result in a larger transportation distance than under a single transportation mode [10]. Since the costs and carbon emissions are mainly from the transportation process, the increased transportation distance (under intermodal transportation situation) may not actually benefit the optimization of the whole transportation systems. Moreover, with the development of the diversified market, the transportation companies are faced with new challenges to meet different preferences as some customers expect a lower cost while some others focus on a fast delivery. In view of the aforementioned observations, the operating decision of IT is in fact a very complex issue. The complex transportation system and resource of freight, the preference of the customers and environmental sustainability (low carbon emissions) ought to be considered to some extent. This paper focuses on the multi-objective multi-sourcing intermodal transportation network design problem (MO_MITNDP) with the aim to find a good trade-off between the transportation cost (TC), maximum flow time (MFT), and carbon emissions (CE). Among them, TC, MFT and CE are associated with economic, efficiency, and societal factors respectively. Furthermore, we have taken in account two kinds of technique constraints, i.e., constraints of transportation modes and constraints of each node’s capacity which are realistic existing to make our model closer to the real-life intermodal transportation situations. Besides, the constraint for each node’s capacity is also a strategy to balance the transportation pressure among the stations, leading to a better sustainability of the transportation system. To formulate the MO_MITNDP, several sourcing places and a single destination are considered. Indeed, this framework is inspired by the well-known complexity of supply chains in the increasing global purchasing and manufacturing environments [11], which has been widely used in many practical transportation systems, such as petrochemical, steel, pharmaceutical industries, etc. The freight transportation company has acted as a freight integrator to transport freights from the sourcing places to the purchasing destination. On the whole, the overall intermodal transportation network is divided into several stages that consist of stations, ports or logistic centers, which are particularly grounded on the geographical characteristics and real intermodal infrastructure construction, such as business types and service radiant scopes. Meanwhile, it is assumed that the freights are irreversibly transported
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from the current stage to the following one without dismounting, which is also a typical assumption in the intermodal transportation problem (Steadieseifi et al., 2014) [12]. The crossing logistics are considered as the business handling capacity constraints for each node of a certain stage when different freights from the corresponding sourcing places are assembled in the same node in a period of time. These constraints are quite close to the real-life situations and significant for balancing the logistics pressures among the stations, as well as reducing the idle transportation resources. In other words, such constraints can disperse the transportation pressure in the station of high-density traffic to the others belong to the same stage, which will further alleviate the city’s traffic congestion and air pollution. The aim of this paper is to determine both the route and mode selection for freight derived from different sourcing places. The main contributions of this paper are listed as follows. Firstly, the MO_MSITNDP is formulated as a multi-objective integer programming model. As for the network structure, the entire intermodal transportation network is divided into several stages according to the geography and social factors. Meanwhile, the capacity constrain of each intermediate node is considered to make the solution reliable in practice and balance the logistics pressure. Secondly, a modified estimation of distribution algorithm (EDA), namely hybrid EDA (HEDA) is proposed to address the mathematical model, which is the first time to apply EDA-based meta-heuristic for MO_MITNDP. Based on the features of MO_MITNDP, we specially design the improvement strategies to enhance the effectiveness and robustness of the proposed method in multi-objective handling capacity. Thirdly, we perform a series of numerous experiments and comparisons based on randomly generated instances to evaluate the effectiveness of HEDA. To further demonstrate the potential application value of HEDA, a case study of Jilin Petrochemical Company (JPC), China, is carried out such that we can obtain the related management insights from the real-world perspectives. The remainder of the paper is organized as follows. Section 2 reviews the relevant literature. In Section 3, the mathematical model of MO_MITNDP is described in detail. Following a detailed description of the proposed HEDA in Section 4, we conduct numerous experiments and comparisons with other algorithms in Section 5, and a case study analysis is performed as well. Finally, conclusions together with future works are presented in Section 6. 2. Literature Review Over the past decades, the IT has attracted many attentions as a cost-effectiveness and environment sustainability alternatives. The existing works can be classified as strategy planning level (Meng and Wang, 2011 [13]; Limbourg and Jourquin, 2009 [14]), tactical planning level (Lam and Gu, 2016 [7]; Verma et al., 2012 [15]), and operational planning level (Francesco et al., 2013 [16]; Bandeira et al., 2009 [17]) according to the decision horizon of the planning problems, and more works can refers to SteadieSeifi et al., 2014 [12]. A majority of the works on ITs consider the cost or time as the single objective and discuss cost-saving or time-efficiency ability compared to the unimodal freight transportation (Bouchery and Fransoo, 2014 [3]). However, the published works regarding cost and time as the bi-objective optimization problem or multiple objectives are limited. Yang et al., 2011 [18] presented a multi-objective optimization model considering the total cost, transportation time and time variability to elevate the competitiveness of 36 candidate routes from China to Indian Ocean. Resat and Turkay, 2015 [19] established a bi-objective mixed-integer programming model with minimizing the total cost and time to determine the transportation modes in the designed intermodal network. Domuta et al., 2012 [20] proposed a modified bi-objective Martines’ Algorithm to find the optimal route in the intermodal transportation network with the time window and the objective is to minimize the travel time and cost. Lam and Gu, 2016 [7] formulated the port hinterland intermodal transportation network design as a bi-objective optimization problem with minimizing the transit cost and time. In view of the green logistics, carbon emissions have been an increasing concern in the transportation sector [7]. Nevertheless, few current studies consider carbon emissions in the field of
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intermodal transportation network. In order to estimate the trade-offs between the transportation cost and carbon emission in the intermodal network and show the modification of the mode and route choice with the changing of the trade-offs, Namseok et al., 2009 [21] developed a decision-support method using the multi-objective optimization. Rudi et al., 2016 [22] established a multi-commodity intermodal network flow model considering the in-transit inventory to make route choices. The carbon emission factor is considered in the objective function which is to minimize the total truckload with the transportation cost, time and carbon emission criteria. Demir et al., 2015 [23] used the sample average approximation method to solve the stochastic intermodal network design problem. The method is applied to provide solutions with the different combination of weights corresponding to the direct transportation, carbon emission-related and time-related cost in the objective function. Considering the cost and carbon emission optimization situations, Bouchery and Fransoo, 2014 [3] presented an intermodal network optimization model to provide decisions for both the terminal location and allocation while Lam and Gu, 2016 [7] studied the carbon emissions in the green intermodal network as a constraint with variable levels. In fact, comparing with single objective, multi-objective optimization can made trade-offs for conflict sub-objectives to satisfy practical fields, which has received wider applications in decision making (Zheng et al., 2015) [2]. In addition, the increasing concerns on the environmental sustainability have promoted it a hotspot to reduce the carbon emission in transportation sector (Lee et al., 2017) [24]. In that regard, The MO_MITNDP considers the cost and time as the objectives, which could help to provide flexible solutions for the decision makers to satisfy the diversified market. The generation of optimal routes is a crucial part of the intermodal transportation network and efficient routes can achieve great resources saving and improve the service quality (Kılıç and Gök, 2014) [25]. According to Verma and Verter, 2010 [26], the main contribution of rail-truck intermodal transportation is the reliability embodied in on-time delivery. The authors make the first attempt to develop a framework of rail-truck intermodal transportation for hazardous goods, in which the route selection is driven by the lead time from diversified customers. Considering the cooperation of inland terminals simulated by a discrete event model, An et al., 2010 [27] designed a service network for the intermodal barge transportation to select the best route with the goal to saving cost. Similarly, Braekers et al., 2013 [28] presented a decision support model for intermodal barge transportation network which involves a major seaport and several hinterland ports to determine the optimal shipping routes aiming to improving the overall profits. Ishfaq and Sox, 2012 [29] integrated an operation model and a location-allocation model for intermodal hub. In their designed hub logistics network, arcs that represent different transportation modes are alternative to increase cost benefits. In this paper, the intermediate nodes in the network are the first time to be divided into several stages grounded on the geographical characteristic and real intermodal infrastructure construction. The ITs faced by the freight integrator is generally a NP-hard problem in which the routes selection, modes combination, terminal location or container scheduling are involved (Li et al., 2015) [30] and huge computational efforts are needed. The meta-heuristic algorithms have an outstanding performance in the combinatorial optimization and have been successfully used in the intermodal freight transportation optimization. Kılıç and Gök, 2014 [25] formulated the problem as an initial route programming for urban transportation network, and a route generation algorithm is designed to find solutions with objectives reflecting the interests of stakeholders. To handle the inland container transportation problem considering hard time constraints, Sterzik and Kopfer, 2013 [31] established a comprehensive model which contains vehicle routing and empty repositioning and a Tabu Search heuristic is used to minimizing the total operation time. Nossack and Pesch, 2013 [32] addressed a truck scheduling problem with time window in intermodal container transportation by a two-stage heuristic. The first stage is route construction heuristic which determines a feasible initial solution for the problem; the second stage is route improvement heuristic which improves solution quality by a neighborhood operator. Genetic algorithm is used in Zhang et al., 2015 [33] to effectively save time and cost.
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optimization problems, there is no research works on EDA for ITs. Essentially, using EDA-based methods to solve MO_MSITNDP is equivalent to confirming the matching probability of the transportation modes and intermediate nodes associated with given freight. Finally, the optimal or Sustainability 2017, 9, 1133 relationship will be found. In this regard, this paper engages in raising the EDA5 of 24 approximate optimal based solution method for dealing with MO_MSITNDP. As a novel swarm intelligence algorithm based on probability model, EDA has been a research 3. Problem Description and Model Formulation hotspot in the area of intelligent computing (Wang et al., 2012 [34]; Wang et al., 2016 [35]). Contrast with genetic algorithm, there is no mutation and crossover operators for EDA, whereas a competitive 3.1. Formulation of Proposed Mathematical Model mechanism is adopted in EDA to estimate the probability model according to excellent individuals considers multiple the sourcing places and can a single purchasing destination. Denote foundMO_MSITNDP in the search process. Thereafter, new population be generated through sampling the N the total number of sourcing places and M the total number of stages, then the scale of estimated probability model. Although EDA has been successfully used in various combinatorial M . Asworks MO_MSITNDP can be defined shownoninEDA Figure there are totallyusing M + 1EDA-based predefined optimization problems, there isas no N research for1,ITs. Essentially, stage 0, stage 1,..., stage M stage 0 ) in the network, where is a of virtual transportation stages (i.e., methods to solve MO_MSITNDP is equivalent to confirming the matching probability the transportation modes and intermediate nodes associated freight. Finally, the optimal point without intermediate nodes. The transport request A1with deriving from the corresponding , ..., Agiven N or approximate will beby found. In this regard, this paper the 1,..., N relationship sourcing placesoptimal will be responded transporting the freights fromengages segmentin1raising to M (i.e., EDA-based method for dealing with MO_MSITNDP. destination solution B). During the transportation process, the transport requests A (i 1,...N ) are processed i
independently without dismounting or consolidation. In this case, there exist crossing logistics in the 3. Problem Description and Model Formulation intermediate nodes owing to the share of intermodal terminal, and the capacity constraint of each 3.1. Formulation of Proposed All Mathematical Model node must be considered. the freights must be irreversibly transported from the stage 1 to M in sequence. The changes of transportation modes only occur on the stages rather than transportation MO_MSITNDP considers multiple sourcing places and a single purchasing destination. Denote N process. Next, we will propose the multi-objective integer programming model of MO_MSITNDP. the total number of sourcing places and M the total number of stages, then the scale of MO_MSITNDP In our model, the objective functions to be minimized consist of the total cost (TTC) and can be defined as N × M. As shown in Figure 1, there are totally M + 1 predefined transportation stages maximum flow time (MFT). Furthermore, the TTC includes three parts, i.e., transportation cost, (i.e., stage 0, stage 1, . . . , stage M) in the network, where stage 0 is a virtual point without intermediate transfer cost of transportation mode and carbon emissions cost, thus our solutions can reflect the nodes. The transport request A1 , . . . , A N deriving from the corresponding sourcing places 1, . . . , N economic benefit, time efficiency and environmental sustainability (society) criteria. The MFT is the will be responded by transporting the freights from segment 1 to M (i.e., destination B). During the maximum completion time of all freights, so a lower MFT indicates a higher efficiency of the whole transportation process, the transport requests Ai (i = 1, . . . N ) are processed independently without systems. Accordingly, our ultimate objective is to find a good trade-off among the economy, efficient dismounting or consolidation. In this case, there exist crossing logistics in the intermediate nodes and sustainability sub-objectives, and these three criteria are usually in conflict with each other. The owing to the share of intermodal terminal, and the capacity constraint of each node must be considered. optimization of IT is essentially to provide a satisfied solution for the intermodal transportation All the freights must be irreversibly transported from the stage 1 to M in sequence. The changes company with reducing the transportation cost, transportation time and carbon emissions of transportation modes only occur on the stages rather than transportation process. Next, we will simultaneously. propose the multi-objective integer programming model of MO_MSITNDP.
Figure Figure 1. 1. Structure Structure of of multi-sourcing multi-sourcing intermodal intermodal transportation transportationnetwork. network.
In our model, the objective functions to be minimized consist of the total cost (TTC) and maximum flow time (MFT). Furthermore, the TTC includes three parts, i.e., transportation cost, transfer cost of transportation mode and carbon emissions cost, thus our solutions can reflect the economic benefit, time efficiency and environmental sustainability (society) criteria. The MFT is the maximum completion time of all freights, so a lower MFT indicates a higher efficiency of the whole systems. Accordingly,
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our ultimate objective is to find a good trade-off among the economy, efficient and sustainability sub-objectives, and these three criteria are usually in conflict with each other. The optimization of IT is essentially to provide a satisfied solution for the intermodal transportation company with reducing the transportation cost, transportation time and carbon emissions simultaneously. Parameters: M: Qi : Rj : R0 : RM : Sj :
An infinite number; Amount of freight from sourcing place i, i ∈ {1, . . . , N }; Set of available nodes of stage j, j ∈ {1, . . . , M − 1}; Available nodes of stage 0 and | R0 |=N; Available node of stage M and | R M |=1; Set of available transportation modes of stage j, j ∈ {1, . . . , M}; The n handling for the kth node of stage j, j ∈ {1, . . . , M − 1} , capacity o k ∈ 1, . . . , R j , W10 , W20 , . . . , WN0 , and W1M = M;
Wkj :
Carbon emission cost of freight i from the lth node in stage j − 1 to the hth node in stage j under the transportation mode w, i ∈ {1, . . . , N }, w ∈ S j , l ∈ R j−1 , h ∈ R j ; Transportation cost of freight i from the lth node in stage j − 1 to the hth node in stage j under the transportation mode w, i ∈ {1, . . . , N }, w ∈ S j , l ∈ R j−1 , h ∈ R j ; Transportation time of freight i from the lth node in stage j − 1 to the hth node in stage j under the transportation mode w, i ∈ {1, . . . , N }, j ∈ {1, . . . , M }, w ∈ S j , l ∈ R j −1 , h ∈ R j ; Switch cost of freight i from transportation mode w to v for the kth node of stage j, i ∈ {1, . . . , N }, j ∈ {1, . . . , M − 1}, w ∈ S j , v ∈ S j+1 , k = 1, . . . , R j ;
eiw ( j − 1, j, l, h) : Ciw ( j − 1, j, l, h) : Tiw ( j − 1, j, l, h) : ξ iw,v ( j, k) :
Switch time of freight i from transportation mode w to v for the kth of stage node j, i ∈ {1, . . . , N }, j ∈ {1, . . . , M − 1}, w ∈ S j , v ∈ S j+1 , k = 1, . . . , R j ; and
τiw,v ( j, k) : ( U w ( j − 1, j)
=
1if transportation mode w exists between stage j − 1 and stage j . 0 otherwise
Decision variables: 1
if transportation mode w is selected for freight i from the lth tnode in stage j − 1 to the hth node in stage j ; 0 otherwise 1 if transportation mode is transfered from w to v yiw,v ( j, k) = ; for freight i in the kth node of stage j 0 otherwise stij the start time at the stage j for freight i; and ctij the leave time at the stage j for freight i. xiw ( j − 1, j, l, h) =
Based on the aforementioned definitions, we detail the mathematical model of MO_MSITNDP as follows. !! w ( j − 1, j, l, h ) · x w ( j − 1, j, l, h ) C i i ∑ ∑ ∑ ∑ w w + e ( j − 1, j, l, h ) · x ( j − 1, j, l, h ) j∈{1,...,M } w∈S j l ∈ R j−1 h∈ R j i i min f 1 = ; (1) ∑ ∑ w,v w,v + ξ ( j, k ) · y ( j, k ) i ∈{1,...,N } ∑ ∑ ∑ ∑ i i k∈ R w∈S v∈S j∈{1,...,M −1}
min f 2 = max
∑
∑
j +1
j
j
∑
j∈{1,...,M } w∈S j l ∈ R j−1
i∈ N +
∑
∑
∑ Tiw ( j − 1, j, l, h) · xiw ( j − 1, j, l, h) h∈ R
∑
j
∑
j∈{1,...,M −1} k ∈ R j w∈S j v∈S j+1
τiw,v ( j, k) · yiw,v ( j, k )
;
(2)
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s. t.
∑
∑ ∑ ∑
j∈{1,...,M } w∈S j l ∈ R j−1 h∈ R j
∑ ∑ ∑
k ∈ R j w ∈ S j v ∈ S j +1
∑
∑ ∑
xiw ( j − 1, j, l, h) = 1; ∀ i ∈ {1, . . . , N };
yiw,v ( j, k) = 1; ∀ i ∈ {1, . . . , N }, j ∈ {1, . . . , M − 1};
i ∈{1,...,N } w∈S j l ∈ R j−1
xiw ( j − 1, j, l, h) Qi ≤ Whj ; ∀ j ∈ {1, . . . , M − 1}, h ∈ R j ;
xiw ( j, j + 1, g, l ) + xiv ( j + 1, j + 2, l, h) ≥ 2yiw,v ( j + 1, k);∀i ∈ {1, . . . , N }, j ∈ {0, . . . , M − 2}, k ∈ R j +1 , w ∈ S j +1 , v ∈ S j +2 ; xiw ( j, j + 1, l, h) ≤ U w ( j, j + 1); ∀i ∈ {1, . . . , N }; j ∈ {0, . . . , M − 1} , w ∈ S j+1 ; stij = cti,j−1 + ∑
∑
∑ Tkw ( j − 1, j, l, h) xiw ( j − 1, j, l, h); ∀i ∈ {1, . . . , N }, j ∈ {1, . . . , M},k ∈ R j ;
w ∈ S j l ∈ R j −1 h ∈ R j
ctij = stij +
∑ ∑ ∑
k ∈ R j w ∈ S j v ∈ S j +1
τiw,v ( j, l ) · yiw,v ( j, k ); ∀i ∈ {1, . . . , N }, j ∈ {1, . . . , M − 1};
(3) (4) (5)
(6) (7) (8) (9)
stij ≥ cti,j−1 ; ∀i ∈ {1, . . . , N }, j ∈ {1, . . . , M };
(10)
xiw ( j − 1, j, l, h) ∈ {0, 1}; ∀i ∈ {1, . . . , N }, j ∈ {1, . . . , M }, l ∈ R j−1 , h ∈ R j , w ∈ S j ;
(11)
yiw,v ( j, k) ∈ {0, 1}; ∀i ∈ {1, . . . , N }, j ∈ {1, . . . , M − 1}, k ∈ R j , w ∈ S j , v ∈ S j+1 ;
(12)
zijk ∈ {0, 1}; ∀i ∈ {1, . . . , N }, j ∈ {0, . . . , M − 1}, k ∈ R j ;
(13)
stij ≥ 0;∀i ∈ {1, . . . , N }, j ∈ {1, . . . , M};
(14)
ctij ≥ 0;∀i ∈ {1, . . . , N }, j ∈ {0, . . . , M − 1}.
(15)
Equations (1) and (2) are the objective functions. Equation (1) is to minimize the TTC which includes transportation cost, transfer cost of transportation mode, and carbon emission cost, whereas Equation (2) minimizes the MFT. Constraint (3) ensures that only one transportation mode can be chosen for two adjacent stages. Constraint (4) ensures the mode transferring only can be executed at the intermodal terminal. Constraint (5) represents the capacity restrictions for each node with respect to given stage. Constraint (6) is the transportation continuity constraints for all routes. Constraint (7) is the transportation mode constraints between two adjacent stages associated with given freight and nodes. Equations (8) and (9) are the definitions of arriving times and departing times with regard to given freight, stages, and selected nodes. Constraint (10) implies that the overall structure of MO_MSITNDP is a directed acyclic graph from the time perspective. Constraints (11)–(15) are the ranges of values for variables. 3.2. Modification of Established Model In view of the aforementioned model, the MO_MSITNDP is essentially a bi-objective optimization problem with capacity constraints, i.e., Constraint (5). Thus, we should introduce relevant constraints handling techniques into HEDA to realize its global search. Unlike single objective optimization problem, the traditional penalty function method may be unusable for multi-objective optimization scenario. In addition, since the solution representation of MO_MSITNDP is a matrix which combines transportation modes with intermediate nodes together, the repair-based constraints handling techniques are more likely to result in a lower efficiency of computation, especially for large scale or strong constraints conditions. Owing to such intractability, we apply the multi-objective-based constraints handling approach [36,37], in which the degree of constraint violation is regarded as an additional objective function. In this regard, the Constraints (5) can be transformed to the third
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total to all intermediate objective function, which denotes the violation with regard degree of constraint w min f max x ( j 1, j , l , h ) Q W ,0 = − − (16) (16): h∈R i 3 i hj ; nodes as shown by Equation j∈{1,..., M −1}
i∈{1,..., N } w∈S j l∈R j −1
Afterwards, the modification of the established modelx wcan be1,given as follows: min f 3 = max ( j − j, l, h ) Q − W , 0 ; i ∑ h∈ R j ∑ ∑ ∑ i hj j∈{1,...,M−1} i ∈{1,...,N } w∈S j l ∈ R j−1 min{ f1 , f 2 , f 3 } . j
(16)
(17)
Afterwards, the modification of the established model can be given as follows: where the first objective functions f 1 and f 2 are the same as Equations (1) and (2). Except for
removing Constraint (5), other constraintsmin of this f 2 , f 3 }. version keep unchanged as the previous (17) { f 1 , modified model. this work, HEDA deals with thef 2modified version model in (1) scene the Pareto whereInthe first objective functions f 1 and are the same as Equations andof (2). Except fordominance. removing The criteria and are the ultimate optimization objective associated with the MO_MSITNDP, f f Constraint (5), other constraints of this modified version keep unchanged as the previous model. 1 2 In this HEDA deals with the modified versionf 3model in scene of the the Pareto dominance. whereas thework, additionally introduced objective function is utilized to guide search direction The criteria f and f are the ultimate optimization objective associated with the MO_MSITNDP, 2 1 from the feasibility perspectives. whereas the additionally introduced objective function f 3 is utilized to guide the search direction from the feasibility perspectives. 4. HEDA for MO_MSITNDP 4.4.1. HEDA for Representation MO_MSITNDP Solution
Since the intermediate nodes and transportation modes need to be determined simultaneously, 4.1. Solution Representation
π=[πand =1,..., N; j =1,..,2 M −1) we specially a matrix to represent MO_MSITNDP with the ij ] (itransportation Since thedesign intermediate nodes modes need to be determined simultaneously, we specially a matrix = πij π(i = denotes 1, . . . , N;the j = index 1, . . . , 2M − 1) to represent N ×Mdesign scale , in which the πelement of selected node (if MO_MSITNDP j mod 2 = 0 ) or ij with the scale N × M, in which the element πij denotes the index of selected node (if jmod2 = 0) or transportation mode (if j mod 2 ≠ 0 ) of the ith sourcing place. In addition, since freight i transportation mode (if jmod2 6= 0) of the ith sourcing place. In addition, since freight i (i = 1, . . . , N) N ) need ( i = 1,..., undergo nodes and M segments in turns after departing fromthe its sourcing sourcing need to undergo M to nodes and MMsegments in turns after departing from its sourcing places, places,(i.e., the sourcing places stage 0) and the common destination (i.e., stage M)to arebenot necessary places stage 0) and the(i.e., common destination (i.e., stage M) are not necessary considered. to be considered. Obviously, if j m o d 2 = 0 ∩ π ij ≤ | R j / 2 | , then π ij denotes the selected intermediate Obviously, if jmod2 = 0 ∩ πij ≤ Rd j/2e , then πij denotes the selected intermediate node of freight i in i =− 1,..., ; j = 1,.., 2πM − 1 . Similarly, noded of i in the selected j /21, π ij denotes stage j/2freight 1, .stage . . , N; j = . . , 2M 1. N Similarly, denotes the selected transportation mode e, for i = , . for ij , if , for transportation moded( jof+ 1freight in segment ( j S +1)/2 of freight i in segment )/2e, if ijmod2 6= 0 ∩ πij ≤ . . . , 2M, −for 1. j mio= d 21,≠ . 0. .∩, N; π ij j≤= | S 1, d( j+ 1)/2e ( j + 1) / 2 | Here, weNtake a simple with thea size N ×example M = 4× 5 tothe illustrate representation × Msolution = 4×5 to i = 1,..., ; j = 1,.., 2M −1example . Here, we take simple with size Nthis illustrate this approach. Considering the matrix solution representation approach. Considering the matrix 1 2 πij = 1 2
3 2 4 1 3 1 1 2 4 1 4 2 4 2 2 1 , 2 2 2 2 2 2 4 1 3 1 3 1 3 2 3 2N×(2M−1)=4×9
which indicates that freight 1 selects the transportation mode 1, 2, 1, 1 and 2 from segment 1 to 5 which indicates that freight 1 selects the transportation mode 1, 2, 1, 1 and 2 from segment 1 to 5 (marked by circles) and selects the intermediate node 3, 4, 3 and 1 in terms of stage 1 to 4, and so on. (marked by circles) and selects the intermediate node 3, 4, 3 and 1 in terms of stage 1 to 4, and so on. 4.2.Multi-Objective Multi-ObjectiveHandling HandlingMethod Method 4.2. Tostore storethe thenon-dominated non-dominated solutions solutions (i.e., (i.e., approximate approximate Pareto Pareto set), set), aa Pareto Pareto archive archive (PA) ( PA) isis To introduced.For For a multi-objective optimization with ifKwecriteria, say that introduced. a multi-objective optimization problemproblem with K criteria, say that ifπ1 we dominatesπ 2 (i.e., ) then both of the following conditions should be π dominates π π π (i.e., following conditions should be satisfied simultaneously: satisfied [38] (1) 1 π1 ≺ π2 ) 2then both1 of the 2 fsimultaneously: for k(1)= f1, (2) it at K least one(2)l it ∈ has . . ,least K } that π{1,..., k has = 1,..., {1, .at , and one f ll(∈ that fk (πand 2 ). 1 ) < Kf}l ( π k ( π1 ) ≤ f k ( π2 ), [38] k (π.1.).≤, K, 2 ) , for Let denote the population size of HEDA, Nπi ( gen)(i = 1, 2, . . . , PS) the ith individual in f l (πPS 1 ) < f l (π 2 ) . Let PS denote the population size of HEDA, Νπi ( gen ) (i = 1, 2,..., PS ) the ith current population at generation gen, AC ( gen) the length of PA at generation gen, Pπi ( gen)(i = individual in current population at generation gen, AC ( gen ) the length of PA at generation gen, 1, . . . , AC ( gen))the ith non-dominated solution of PA at generation gen. The newly generated Pπ i ( gen ) (Nπ i = 1,..., AC ( gen )) the ith non-dominated solution of PA at generation gen. The newly individual i ( gen )(i = 1, 2, . . . , PS ) will be added to PA if and only if it can dominate at least generated individual Νπi ( gen ) (i = 1, 2,..., PS ) will be added to
PA if and only if it can dominate at
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one in current PA, PA and, meanwhile, thethe solution that is isdominated leastsolution one solution in current , and, meanwhile, solution that dominatedby bynewly newlygenerated generated individual will be removed from PA. Here, we propose the update procedure of PA as shown in PA as individual will be from . Here, wethe propose thethat update procedure shown in least one solution in removed current PA , and,PA meanwhile, solution is dominated byofnewly generated Figure 2. Figure 2. individual will be removed from PA. Here, we propose the update procedure of PA as shown in Figure 2. Algorithm 1: Update the Pareto archive. := Update 1 to PS do 01 For i1: Algorithm the Pareto archive. For ii:=PS 1 to 1 to doAC(gen) do 0102 For i:= ( gen )) Pπ ii ( gen ) then For iiIf := 1Νπ to iAC (gen ) do 0203
0304 0405 0506 0607 0708 0809 0910 1011 1112 1213 1314 1415 15
PA; PA;
If ΝπΝπ )) )) PπPA ) then , and remove Pπii ( gen ) from the current i ( gen ii ( gen i ( gen For iii:= ii + 1 to AC(gen) do Νπ i ( gen )) PA , and remove Pπ ii ( gen ) from the current ) (Pπ gen) then For iiiIf:= iiΝπ + i1( gen to AC geniii)(do
PA; PA;
If Νπ Remove from the current Pπiiiii ( gen)) then i ( gen ) Pπ End ;//end ifPπii ( gen ) from the current Remove End ; //end End;//endfor if iii End //end if for iii End ; //end Else End //end if Else Continue; //skip to the next loop End ; //end else Continue ; //skip to the next loop End ; //end forelse ii End; //end End ; //end procedure End ; //end for ii
End; //end procedure Figure 2. The procedure of updating the Pareto archive. Figure 2. The procedure of updating the Pareto archive. Figure 2. The procedure of updating the Pareto archive.
4.3. The Proposed Heterogeneous Probability Model and Update Mechanism 4.3. The Proposed Heterogeneous Probability Model and Update Mechanism 4.3. TheWhen Proposed Heterogeneous Probability Modelcombinatorial and Update Mechanism using EDA to solve the discrete optimization problems, the probability When using EDA to solve the discrete combinatorial optimization problems, the probability model is generally described as joint probability distribution function problems, with independent random When using EDA to solve the discrete combinatorial optimization the probability model is generally described as joint probability distribution function awith independent marginal random variables. Considering the inherent features of MO_MSITNDP, heterogeneous model is generally described as joint probability distribution function with independent random variables. Considering the is inherent features MO_MSITNDP, a heterogeneous marginal distribution distribution law (HMDL) forward forofserving as HEDA’s probability model, which is utilized variables. Considering the put inherent features of MO_MSITNDP, a heterogeneous marginal law (HMDL) is put forward for serving as HEDA’s probability model, which is utilized to represent to represent the distribution of promising solution regions. The probability model pr ( gen ) can be distribution law (HMDL) is put forward for serving as HEDA’s probability model, which is utilized the distribution of promising solution regions. The probability model pr( gen) can be formulated as follows: toformulated represent the distribution of promising solution regions. The probability model pr ( gen) can be as follows: pri ( gen ) P i ( gen) Ri ( gen) formulated as follows: i i i i pri (genp)i ( gen P i ()gen) pR1i ij ( gen) p1M ( gen) r11 ( gen) r1 j ( gen) r1M 1 ( gen) 11 i i i i i i ) p ( gen p1M ( gen) r11 ( gen) r1 j ( gen) r1M 1 ( gen) p11 (i gen) 1 j i i i i i ) rrj( gen) rrM 1 ( gen) , ps1 ( gen) psj ( gen) psM ( gen) rr1( gen i i i i i ) rrM ) psM ( gen) rr1 ( gen) rrji ( gen ps1 ( gen) psj ( gen 1 ( gen) , i i i p i i ) r| Ri |, j ( gen ( gen ) pmaxS, ) r| RM|, M 1 ( gen) , M ( gen) r| R1|,1 ( gen ) pmaxS j ( gen j maxS,1 i i pi pi r i ( gen) r| Ri |, M 1 ( gen) ( ) r gen ( ) gen ( gen) pmaxS, ( ) gen R | |,1 maxS M , maxS, 1 | |, R j M j j 1 i ( gen) i ( gen) P R for i 1,..., N , P i ( gen) Ri ( gen)
(18) (18)
i i 1,...,max N , {| S j |} is the maximum value of transportation modes, psj ( gen) is the maxS where for j{1,..., M }
i
where maxS max {| S j |} the maximum value of transportation modes, psj ( gen) is the is mode probability ofj{1,..., transportation to be selected in the jth segment for the iith freight at M} S j is thes maximum where maxS = max value of transportation modes, psj ( gen) is the j∈{1,...,M } i ithj freight probability of transportation mode s to be selected in the the for the jth segment rth ith generation of gen , and rrj ( gen)mode is the stage the at probability transportation s toprobability be selectedofinselecting the jth segment for node the ithinfreight at for generation i
i generation rrjgen ( gen generation gen , and probability the jrth node in stage j for the ith . ) is the freight gen, andatrrj ( gen ) is the probability of selecting theof rthselecting node in stage for the ith freight at generation gen. j 1 j j 1,..., M Since thereexists exists modeconstraint constraint between stage there between stage j − 1 andand j (for j (for = 1, . . . , M), the), gentransportation .transportationmode freightSince at generation w ( j − 1,wj )=0 (for j = 1, . . . , M, w = 1, . . . , maxS). Therefore, the relate probability should be set to 0 if u w 1,..., maxS j 1,..., M theSince relate probability should be set to 0 if =0 (for , Therefore, u ( j 1, jbetween ) M ), there exists transportation mode constraint stage j 1 and j (for j ). 1,..., probability model can be initialized as follows: w the probability model can be initialized as follows: the relate probability should be set to 0 if =0 (for j 1,..., M , w 1,..., maxS ). Therefore,
u ( j 1, j )
the probability model can be initialized as follows:
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Pi ( 0 ) =
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1/|S1 | umaxS (0, 1) ... maxS 1/|S1 | u ( 1, 2 ) m axS ... u 1 (0,1) (0,1) · 1/ | S1 | ... u. .. 1 ... .. m axS ... u (1, 2) 1/ | .S1 | u (1, 2) ... 1/|S1 | umaxS...( M − 1, M ) u 1 ( M 1, M ) 1/ ... | Ru1m|axS ( M1/ 2 | ) 1/ | S1 | |1,RM ··· 1/| R1 | 1/ | R 1/||R2 | 1/ | iR1 | 1/ M | R 2 | R ( 0 ) = ··· . . 1/ | R | 1/ | R | . . 1/ | R M | .. . 2 1 ··· R i (0) 1/| R1 | 1/|R2 |
u1 (0, 1) u1 (1, 2) .. . i P (0) u1 ( M − 1, M)
1/ | R1 | 1/ | R 2 |
1/ | R M |
1/|S2 | ··· 1/|S2 | 1/ | S 2 |. 1/ ·| ·S·M . 1/ | S 2 |. 1/ ·| ·S·M · · · 1/|S2 | 1/ | | 1/ | SM S 1/| R2M | 1/| R M | . .. . 1/| R M |
1/|S M | 1/|S M | .. . , 1/|S M |
| | | m axS M
, maxS× M
(19)
(19)
Probability model update is one of the crucial operators for EDA [32], which is used to accumulate Probability model update is one of the crucial operators for EDA [32], which is used to the historical information of excellent individuals found during the search process. the new accumulate the historical information of excellent individuals found during the Thereafter, search process. populations are generated by sampling the updated probability model. As mentioned above, PA is Thereafter, the new populations are generated by sampling the updated probability model. As dynamically updated generated individuals to theindividuals dominanceaccording rule (see algorithm PAnew mentioned above, by is dynamically updated by according new generated to the in Figure 1), so that PA can be seen as a set of excellent individuals. In view of this, we apply the dominance rule (see algorithm in Figure 1), so that PA can be seen as a set of excellent individuals. non-dominated solutions that located in PA to update the probability model. Denote LR the learning In view of this, we apply the non-dominated solutions that located in PA to update the probability rate model. of HEDA and LR x, ythe thelearning temporary variables, thenx,the probability model update is given in y the rate of HEDA and temporary variables, thenmethod the probability Denote Figure 3. model update method is given in Figure 3. Algorithm 2: Update the probability model. e
e
01
Randomly select an individual [ ij ]N (2M 1) from the current PA;
02 03 04 05
Set x : 1, y : 1
06 07 08 09 10 11
For i:= 1 to N do For j:= 1 to 2M − 1 do If j mod 2 0 then pi e , x ( gen) : pi e , x ( gen) LR ;// concerned with transportation modes ij
ij
Normalize the probability
i psx (gen) for each s 1,..., maxS ;
x : x 1;//the next segment End Else ri e , y ( gen ) : ri e , y ( gen ) LR ;//concerned with intermediate nodes ij
ij
i
12
Normalize the probability rry ( gen) for each r 1,...,| R y | ;
13 14 15 16
y : y 1 ;//the next stage End;//end else End;//end for j End;//end procedure Figure 3. The procedure of updating the probability model.
Figure 3. The procedure of updating the probability model.
4.4. New Population Generation Method
4.4. New Population Generation Method
HEDA generates new population by sampling the updated probability model (i.e., HMDL). In HEDA generates new population by sampling the updated model HMDL). In the the sampling procedure, the new generated individuals should beprobability consistent with the(i.e., HMDL to guide HEDA’s global search and, meanwhile, to keep the population diversity, relevant random sampling procedure, the direction, new generated individuals should be consistent with the HMDL to guide factors may need to direction, be considered. is, we tendtotokeep selectthe a random variable in HMDL with larger HEDA’s global search and,That meanwhile, population diversity, relevant random probability for producing new individuals, whereas ones corresponding to relative smallerwith factors may need to be considered. That is, we tend to select a random variable in HMDL probability alsofor have a chance tonew be selected. In thiswhereas regard, we propose a sampling to approach larger probability producing individuals, ones corresponding relativebased smaller on roulette wheel selection, for which the selection probability of a random variable is proportional
probability also have a chance to be selected. In this regard, we propose a sampling approach based i i i T ), FPmaxS j proportional 1,..., M ) to its probability in HMDL. FP ji selection ( gen ) [FP1,probability j (gen),...,FPmaxS,j +1, j (gen)] ( is on roulette wheel selection, for Denote which the of (agen random variable i i i i to itsthe probability in HMDL. Denotefunction FP j ( gen(MDF) ) =[ FP1,j ), . . . , FPmatrix )]T ,(j = P i),FP distribution of( gen probability at maxS generation j th marginal maxS,j ( gen +1,j ( gengen 1, . . . , M) the jth marginal distribution function (MDF) of probability matrix Pi at generation T
i ( gen ), . . . , FRi gen, FRij ( gen) = [ FR1,j ( gen), FRi| R |+1,j| ( gen)] (j = 1, . . . , M − 1) the jth MDF of | R |,j j
j
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FRij ( gen) [ FR1,i j ( gen),..., FR|iR j |, j ( gen), FR|iR j |1, j| ( gen)]T ( j 1,..., M 1 ) the j th MDF of probability i T i at),..., FRij ( gen)matrix [ FRi jR ( gen FR|iR j |, j ( gen ), FR 1,..., M 1 variables. ( jtemporary ) the j th Then, MDF the of probability probability generation gen, and rnd)]the procedure of |R |x, 1, jy, | ( gen Ri at 1,generation matrix gen, and x , y ,j rnd the temporary variables. Then, the procedure of i calculating MDF and generating new population are shown in Figures 4 and 5. matrix R MDF at generation gen, new and population variables. Then, the procedure of x , y , rnd the calculating and generating aretemporary shown in Figures 4 and 5. calculating MDF and generating new population are shown in Figures 4 and 5. Algorithm 3: Calculate MDF of HMDL. For i:=3:1Calculate to N do MDF of HMDL. 01 Algorithm M do 02 ForFor i:= 1j:=to1Ntodo 01 i For j:= 1 to 02 gen) : 0 ; 03 Set FPM(do 03 04 04 05 05 06 07 06 08 07 08 09 09 10 10 11 11 12 13 12 14 13 14
1, j i
) : 0+1 ; Set FP j ( gen maxS do For k:= 21, to i do maxS +1 For FP k:= i2 to ( gen) : FP ( gen) pi
k, j k 1, j k 1, j ( gen) ;//calculate MDF i i i FP ( gen ) : FP ( gen ) p k , j for k k 1, j k 1, j ( gen) ;//calculate MDF End;//end
End;//end for jfor k End;//end For j:= 1 to for M −j 1 do End;//end i − 1 do ForSet j:= 1FR to M ( gen) : 0 ; 1, j i
Set FR1, j ( gen) : 0 ; For k:= 2 to | Rj | 1 do For k:= i2 to | Rj | 1 i do
FRk , j ( gen) : FRk 1, j ( gen) rki1, j ( gen) ;//calculate MDF FRki , j (for genk) : FRki 1, j ( gen) rki1, j ( gen) ;//calculate MDF End;//end
End;//end for jfor k End;//end End;//end procedure End;//end for j End;//end procedure
Figure 4. The procedure of calculating MDF. Figure 4. The procedure of calculating MDF. Figure 4. The procedure of calculating MDF. Algorithm 4: Generate new population. For h:=4:1Generate to PS do new population. 01 Algorithm For i:= to N 02 For h:= 1 to1 PS dodo 01 x : y : 1 ; Set 03 For i:= 1 to N1,do 02 x : 1, y :−11; do Set For j:= 1 to 2M 04 03 05 For rnd j:= 1: torandom 2M − 1 [0,1) do ; 04 ; rndj :mod random then If 05 2 0[0,1) 06 06 07 07 08 08 09 09 10
11 10 12 11 13 12 14 13 14 15 15 16 16 17 17 18 19 18 20 19 21 20 22 21 23 22 24 23 24
then do If For j mod k:=21 to0 maxS i do For k:= FP 1 toi (maxS If k, x gen) rnd FPk 1, x ( gen) then i i If FPk, x (gen) rnd FPk 1, x ( gen) then Νπh,i, j (gen): k ;//sampling transportation mode Νπ (gen): k transportation mode x :hx,i,j 1;//the next;//sampling segment End;//end x : x if1;//the next segment End;//end for kif End;//end EndEnd;//end for k Else End ElseFor k:= 1 to | R | do j For k:= 1 toi | Rj | do i If FRk , y ( gen) rnd FRk 1, y ( gen) then i i If FRk , y ( gen) rnd FRk 1, y ( gen) then Νπh,i, j (gen): k ;//sampling intermediate node Νπ ): knext ;//sampling intermediate node y :h,iy, j(gen 1 ;//the stage
y : y if 1 ;//the next stage End;//end End;//end for kif End;//end End;//end else End;//end for k End;//end for jelse End;//end End;//end for ifor j End;//end End;//end for hfor i End;//end End;//end for h Figure 5. The procedure of generating new population. Figure 5. The procedure of generating new population. Figure 5. The procedure of generating new population.
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4.5. Problem-Dependent Local Search 4.5. Problem-Dependent Local Search As mentioned above, HEDA may pay little attention to the local search during its iterative iterative As mentioned above,be HEDA pay little attention to the local search during iterative process. Hence, it would highlymay advisable to embed problem-dependent local searchitsinto HEDA process. Hence, Hence, ititwould wouldbe behighly highlyadvisable advisabletotoembed embedproblem-dependent problem-dependent local search into HEDA process. local search into HEDA to to exploit more promising solutions in relatively narrow regions of solution space. Since the solution to exploit more promising solutions in relatively narrow regions of solution space. Since the solution exploit more promising solutions in relatively narrow regions of solution space. Since the solution representation of MO_MSITNDP is a N M matrix, traditional neighborhood structures, such as M representation of MO_MSITNDP is a NN× M matrix, traditional neighborhood structures, such as Insert, Swap, Inverse, etc. [39], may be unsuitable. In this work, we propose a new neighborhood be unsuitable. unsuitable. In this work, we propose a new neighborhood neighborhood Insert, Swap, Inverse, etc. [39], may be structure named partially interchanged neighborhood (PIN), in which two random positions namedpartially partiallyinterchanged interchanged neighborhood (PIN), in which two random positions structure named neighborhood (PIN), in which two random positions c1 , c2 ∈ c1 , c2 {1,..., 2 M 1} , c1 c2 are selected first and then we interchange the partial elements between , are selected first and then we interchange the partial elements between , c {1,..., 2 M 1} c c . . . , 2M − 1 , c < c are selected first and then we interchange the partial elements between c1 {c1, } 1 1 2 12 2 c 1 and c 2 associated with two randomly selected r1 and r 2 , r1 , r2 {1,..., N } , r1 r2 . The and c associated with two randomly selected r and r , r , r ∈ 1, . . . , N , r 6 = r . The mechanism { r 2 , r1 , r}2 1{1,...,2N } , r1 r2 . The 2 2 r11 and 2 c 1 and c 2 associated with two randomly 1selected mechanism of PIN can be illustrated by a simple example, as shown Figure 6. of PIN can be illustrated by a simple example, as shown in Figure 6. ininFigure mechanism of PIN can be illustrated by a simple example, as shown 6.
Figure6.6.The Theschematic schematicdiagram diagramofofpartially partiallyinterchanged interchangedneighborhood neighborhood(PIN). (PIN). Figure Figure 6. The schematic diagram of partially interchanged neighborhood (PIN).
Thereafter, we perform the PIN-based local search for all non-dominated solutions in PA, and Thereafter, we perform the PIN-based local search for all non-dominated solutions in PA, and meanwhile the first-move rule is adopted in order to avoid excessive exploitation. That is, once a meanwhile the first-move rule is adopted adopted in in order order to to avoid avoid excessive excessive exploitation. exploitation. That is, once a relatively better solution is found from the neighborhood of current solution, the local search will be relatively better solution is found from the neighborhood of current solution, the local search will be terminated. Let β ( ) denote a neighborhood solution associated with performing PIN operator terminated. Let Let ββ= `( denoteaaneighborhood neighborhoodsolution solutionassociated associatedwith withperforming performing PIN operator (π )) denote cnt one time on , the temporary variables. Then, the proposed local search operator giveninin one time on on π, cnt the temporary variables. Then, the proposed local search operator is isgiven one time , cnt the temporary variables. Then, the proposed local search operator is given in Figure 7. Figure Figure 7. 7. Algorithm 5: PIN-based local search. Algorithm 5: PIN-based local search. 01 For i:= 1 to AC ( gen ) do 01 For i:= 1 to AC ( gen ) do cnt : 0 ; 02 cnt : 0 ; 02 Repeat 03 Repeat 03 β : ( Pπ i ( gen )) ;//construct the neighborhood solution via PIN operator 04 β : ( Pπ i ( gen )) ;//construct the neighborhood solution via PIN operator 04 cnt : cnt 1 ; 05 cnt : cnt 1 ; 05 Until ( cnt 2M 1 or β Pπi ( gen) ) 06 Until ( cnt 2M 1 or β Pπi ( gen) ) 06 β Pπ i ( gen ) ;//replace the current solution with its neighborhood solution 07 β Pπ i ( gen ) ;//replace the current solution with its neighborhood solution 07 08 End;//end for i 08 End;//end for i 09 Update the current PA; 09 Update the current PA; Figure 7. The procedure of PIN-based local search. Figure 7. The procedure of PIN-based local search. Figure 7. The procedure of PIN-based local search.
4.6. Procedure of HEDA 4.6. Procedure of HEDA 4.6. Procedure of HEDA Let maxGen denote the maximum number of iteration for HEDA. Based on aforementioned maxGen Let denote the maximum number of iteration for HEDA. Based on aforementioned subsections, the procedure HEDA is shown Figure 8. for HEDA. Based on aforementioned Let maxGen denote theofmaximum numberinof iteration subsections, the procedure of HEDA is shown in Figure 8. subsections, the procedure of HEDA is shown in Figure 8.
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Algorithm 6: HEDA. 01 For gen:= 1 to maxGen do If gen=1 then 02 Randomly initialize the population Νπ i ( gen 1) , for i 1, 2,..., PS ; 03
Construct the initial PA Pπ i ( gen 1) , for i 1,..., AC ( gen 1) ; Initialize the probability model pri ( gen 1) , for i 1,..., N ;
04 05 06 07 08
End Else Update the probability model pri ( gen ) , for i 1,..., N ;// Algorithm 2
09
Generate new population Νπi ( gen ) , for i 1, 2,..., PS ;// Algorithm 4
10 11 12 13 14
Update the PA Pπ i ( gen ) , for i 1,..., AC ( gen ) ;// Algorithm 1 Perform the PIN-dependent local search;// Algorithm 5 End;//end else End;//end for gen Output Pπi (maxGen) , for i 1,..., AC ( maxGen ) ; Figure 8. The procedure of HEDA. Figure 8. The procedure of HEDA.
5. Computational Results and Comparisons 5. Computational Results and Comparisons To evaluate the performance of HEDA, we first perform computational experiments and To evaluate the performance of HEDA, we first perform computational experiments and comparisons based on randomly generated instances. Then, a real-life case study of Jilin comparisons based on randomly generated instances. Then, a real-life case study of Jilin Petrochemical Petrochemical Company (JPC), China, is carried out to demonstrate the application value of the Company (JPC), China, is carried out to demonstrate the application value of the proposed HEDA. proposed HEDA. 5.1. Experimental Setup 5.1. Experimental Setup Since there is no benchmark data for MO_MSITNDP, we consider fifteen different combinations Since there is no benchmark data for MO_MSITNDP, we consider fifteen different combinations of N × M that include 5 × 3, 10 × {3, 4, 5}, 20 × {4, 5, 6, 7}, 30 × {6, 7, 8, 9}, 40 × {9, 11} and 50 × 13 of N M that include 53 , 10 {3, 4, 5} , 20 {4, 5, 6, 7} , 30 {6, 7,8, 9} , 40 {9,11} and 5013 for for generating test instances. For each combination, we randomly generate 10 instances such that generating testof instances. For each we randomly generate are 10 instances such that(1) there there is a total 150 instances. Thecombination, generation rules of these instances listed as follows: the is a totalof ofthe 150ith instances. The generation rules of1,these instances are listed as follows:nodes (1) theinamount amount freight Q ∈ [ 100, 500 ] , for i = . . . , N; (2) the number of available the jth i i 1,..., N of the i th freight , for ; (2) the number of available nodes in the j th stage Q [100,500] stage R j ∈ [1, 6] , fori j = 2, . . . , M − 1, obviously | R0 |= N and | R M |= 1 ; (3) the transportation mode w(j − j 1,2,..., M j1= , obviously theassigning transportation mode constraint |RM | 1 ; (3) by | R j | [1, 6] , ufor constraint j), for 1, . . . , M, is|Rrandomly waterway, railway and 0 | N and established w road each with 1; (4) the established carbon emission cost ei ( jwaterway, − 1, j, l, h) railway ∈ [1, 100and ], transportation by assigning road in each for j 1,..., M ,0isorrandomly u w ( j in 1, j ) , segment w ( j − 1, j, l, h ) ∈ [1, 100], transportation time T w ( j − 1, j, l, h ) ∈ [1, 10], for i = 1, . . . , N, j = cost C w i i e ( j 1, j , l , h ) [1,1 00 ] , transportation cost segment with 0 or 1; (4) the carbon emission cost i 1, .w. . , M, w = S j , l ∈ R j−1 ; (5) the switch cost ξ iw,v ( j, k ) ∈ [1, 10], switch time τiw,v ( j, k) ∈ [1, 3], w C i ( j 1, j , l , h ) [1,100] , transportation time Ti ( j 1, j , l , h ) [1,10] , for i 1,..., N , j 1,..., M , w S j for i = 1, . . . , N, j = 1, . . . , M − 1, w ∈ S j , v ∈ S j+1 , ]; and (6) the handling capacity is calculated as w,v wv , N j, k10),W k)1, [1,10] [1,3] 1,...,WN1M, · R j , r ∈ [0.1, 0.5], ifor( jj,= . . . , M, − 1, k = time 1, . . . , R , WN0i and ,Wkj (5) switch cost switch l =R∑ ; Q 20 , . ., . for i /rthe i j (, W j i1=1
are set to be infinite values. N j 1,..., M 1 , w S j , v S j 1 , ]; and (6) the handling capacity is calculated as Wkj i 1 Qi / r | R j | We independently run all algorithms 20 times, and the results are all based on the average level of r [0.1,0.5]with j 1,..., , instances , forrespect , k 1,N..., areMAX, set toAVG be and infinite WNthe W1M R j combinations. | , W10 , W20 ,...,In 0 and 10 toM the 1related ×| M results, MIN, SD represent values. the minimum, maximum, average and standard deviation, respectively, corresponding to the related performancerun metrics. The unique terminated condition for all based algorithms the maximal We independently all algorithms 20 times, and the results are on theisaverage level evaluation times of objective functions, which is set to 1000 × N × M. All algorithms are coded in of 10 instances with respect to the related N M combinations. In the results, MIN, MAX, AVG C++ and in Studio 2008 environment, andand all the simulations are conducted on acorresponding PC with Intel SDMicrosoft representVisual the minimum, maximum, average standard deviation, respectively, Core-i5 3.30 GHz processormetrics. with 4 GB to the related performance Thememory. unique terminated condition for all algorithms is the maximal
evaluation times of objective functions, which is set to 1000 N M . All algorithms are coded in C++ 5.2. Performance Metrics in Microsoft Visual Studio 2008 environment, and all the simulations are conducted on a PC with systematically the performance of HEDA, three performance metrics [40] are adopted Intel To Core-i5 3.30 GHz verify processor with 4 GB memory. as follows:
Sustainability 2017, 9, 1133 Sustainability 2017, 9, 1133 Sustainability 9,9,1133 Sustainability 2017, 9, 1133 Sustainability 2017, 9, 1133 Sustainability 2017, 1133 Sustainability 2017, 9, 2017, 1133 Sustainability 2017, 9, 1133 Sustainability 2017, 9, 1133
14 of 24 14 of 24 14 of of24 2424 1424 of 241414of 14 14 of of 24 14 of 24
Sustainability 2017, 9, 1133 5.2.Metrics Performance Metrics 5.2. Performance 5.2. Performance 5.2. Performance Metrics 5.2. Performance Metrics 5.2. Performance Metrics 5.2. Performance MetricsMetrics 5.2. Performance Metrics 5.2. Performance Metrics Sustainability 2017, 9, 1133 of 24 To systematically verify the of performance of performance HEDA, three performance metrics [40] are 14 adopted To systematically verify the performance HEDA, three metrics [40] are adopted 5.2. Performance Metrics To systematically verify the performance ofof HEDA, three performance metrics [40] are adopted To systematically verify the performance of HEDA, three performance metrics [40] are adopted systematically verify the performance of HEDA, three performance metrics [40] are adopted To systematically verify the performance HEDA, three performance metrics [40] are adopted To systematically verify the performance of HEDA, three performance metrics [40] are adopted ToTo systematically verify the performance of HEDA, three performance metrics [40] are adopted To systematically verify the performance of HEDA, three performance metrics [40] are adopted as follows: as follows: asasas follows: follows: follows: follows: as follows: asas follows: To systematically verify the performance of HEDA, three as follows: (1) Overall Overall non-dominated solutions number (ONSN) (1) Overall non-dominated solutions number (ONSN) (1) non-dominated solutions number (ONSN) as follows: (1) Overall non-dominated solutions number (ONSN) (1) Overall non-dominated solutions number (ONSN) solutions number (ONSN) (1) Overall non-dominated solutions number (ONSN) (1)Overall Overall non-dominated solutions number (ONSN) (1)(1) Overall non-dominated solutions number (ONSN) (1)non-dominated Overall non-dominated solutions number (ONSN) Sustainability 2017, 9, 1133 Denote is the union of non-dominated with regard to (ONSN) Denote = 1 solutions withsolutions regardsolutions to = Lis2 , the ..., union L of non-dominated , ..., 1 2 (1) Overall non-dominated number Denote isunion the union of non-dominated solutions with tototo = the Denote is the union of non-dominated solutions with regard Denote is of non-dominated solutions with regard toregard Denote is the union of non-dominated solutions with regard == ,..., ..., = , ..., Denote is the of non-dominated solutions with regard to = , ..., Denote is the union of non-dominated solutions with regard to = , ..., 1 2 2,L Lunion = , ..., 1 2 L 1 2 Denote is the union of non-dominated solutions with regard to 1 L 1 2 L , ..., 1 2 L Denote is 1 L the union of non-dominated solutions with regard to , and then is of thenon-dominated number of non-dominated in l 1,...,then l ,..., L ONSN the (number solutions in solutions Algorithm 1,..., lAlgorithm ,..., L , and ( )2 isONSN l l )is Metrics l5.2. , and then the number of non-dominated solutions in Performance Algorithm 1,..., l ,..., L ONSN ( ) , and then is the number of non-dominated solutions in Denote is the union of non , and then is the number of non-dominated solutions in Algorithm 1,..., l ,..., L ONSN ( ) Algorithm 1,..., l ,..., L ONSN ( ) , and then is the number of non-dominated solutions in = , ..., Algorithm 1,..., l,,..., ONSN ( and is the number of non-dominated solutions Algorithm 1,...,Ll ,..., L1,..., ONSN l l )l is ll l , 1, and then isONSN non-dominated Algorithm 1,..., l ,..., ( l solutions 1 in solutions 2inin , and then is of the number of non-dominated Algorithm l ,..., (number l )the l inL Algorithm .not .,.by l, . L. .Lthen , ONSN L, then number of(20). non-dominated solutions l )(l l ) the which are dominated by solutions in as shown in Equation (20). The larger value ofl which are not dominated solutions in as shown in Equation The larger the value ofl the which which are not dominated by solutions in asin shown inin Equation (20). The larger value which are not dominated by solutions in as shown in Equation The larger the value which not dominated by solutions in as shown Equation (20). The larger the value of which are not dominated by solutions in shown Equation (20). The larger the value ,the and then number which are not dominated solutions in as shown in Equation (20). The larger the value of Algorithm 1,..., llarger ,..., L(20). ONSN (of ) of which areare not dominated by solutions in shown in Equation (20). The larger the value ofthe To systematically verify the performance of HEDA, three performance [40 which are not dominated by solutions in as shown in Equation (20). The larger the value ofis themetrics l of as are not dominated byby solutions in as shown inas Equation (20). The value of is, the better of performance l. (l ) performance the better Algorithm l.of Algorithm ONSN (l ) is,ONSN is, the better performance of Algorithm l . ONSN ( ) is, the better performance of Algorithm l . which are not dominated by solutions in as shown in is, the better performance of Algorithm l . as follows: ONSN ( ) ONSN ( ) is, the better performance of Algorithm l . ONSN ( is, better performance of Algorithm l. (l the better ofof Algorithm l. l. l l )l performance is, better performance Algorithm ONSNONSN (l )is, ONSN (the l )the l ) is, the better performance of Algorithm l. (x )Overall l | | x } | . better(ONSN) O N SN ( l ) | O NlSN \ {(1) y l \ {non-dominated : l y| y xONSN | . (20) performance of Algorithm l. (20) (: l y) is,x }the solutions SN )lx)\ll \|{ x\yx{| | yy|:y .| .. SN ((\x )| || y NO SN (SN )NO |N {l x (20) (20)(20) (20) O N SN (l{ | . | (O ) (\| x :x x. :}:y| .y:yxx}}x| .|}number l l l (20) O NOSN (N | y }y |} (20) l y\l { l :|l lN l l{) xl xy (20) l ) l | O lSN l l \ { l | y : y x } | (20) Sustainability 2017, 9, 1133 N SN ( ) | l \ { x l solutions | y w Denote(RNDS) of l non-dominated = 1 2 , ..., L is the Ounion Ratio of non-dominated solutions (2) Ratio of (2) non-dominated solutions (RNDS) (2) Ratio of non-dominated solutions (RNDS) (2) Ratio of non-dominated solutions (RNDS) (2) Ratio of non-dominated solutions (RNDS) (2) Ratio of non-dominated solutions (RNDS) (2)Ratio Ratio of non-dominated solutions (RNDS) (2) Ratio of non-dominated solutions (RNDS) Sustainability 2017, 9, 1133 14 of 24 (2) of non-dominated solutions (RNDS) (2) Ratio of non-dominatedAlgorithm solutions1,..., (RNDS) , and then ONSN (l ) is the number of non-dominated solut l ,..., LMetrics 5.2.not Performance Denote the ratio which of solutions which are notbydominated by solutions in solutions as ratio of solutions are dominated solutions in shown as , shown , in ty 2017, 9, 1133Denote theDenote 14 of 24in (2) Ratio of non-dominated (RNDS) the ratio of solutions which are not dominated by solutions ,as asasas Denote the ratio solutions which are not dominated solutions ,,shown shown Denote the ratio ofsolutions solutions which arenot not dominated by solutions in , in shown Denote the ratio ofof which are not dominated byby solutions in , shown Denote the ratio of solutions which are not dominated by solutions , shown as which are dominated by solutions in as shown in Equation Denote the ratio of which are dominated by solutions in , shown as Denote the ratio ofsolutions solutions which are not dominated by solutions in shown as (20). The large Denote the ratio of solutions which are not dominated by solutions in , shown as following 5.2. Performance Metrics To systematically verify performance of HEDA, of three performance metrics is, the better of performance Algorithm following (21). higher the value (24 l ) the the better performance Algorithm following Equation (21).Equation The higher theThe value of RNDS (l )of is,RNDS 14 of is, the better performance ofofof Algorithm following Equation (21). The higher the value of )l )Denote is, the better performance Algorithm following Equation (21). The higher the value of the ratio solutions which are not dominate is, (the performance of Algorithm following Equation (21). TheThe higher the value of (( RNDS (the )better is, the better performance following Equation (21). The higher the value of RNDS (performance is, the performance of following Equation (21). higher the value of RNDS (RNDS is, of Algorithm l. Algorithm l better ONSN ( is,RNDS better performance of Algorithm Equation (21). The the value RNDS ( l ))better l is, the better performance ofAlgorithm Algorithm following Equation (21). The higher the RNDS l )the l ) value rmancelfollowing Metrics l )of l performance asof follows: Equation (21). Thehigher higher the value of is, the better Algorithm l. . To systematically verify the performance of HEDA, three performance metrics [40]ofare adoptedl. l . l. ll.. following Equation (21). The higher the value of RNDS (l ) l. l. l. systematically as follows: verify the performance of HEDA, three performance metrics [40] ON SN (are ladopted ) | \ { x l |(ONSN) y : y x } | . (1) Overall non-dominated . S ( ) | \ { x | y x } | / | l number R N D S ( l ) | R N D \ { x | y : y x } | /. | l: |y.solutions (21) (21) l l l l | l l l (21) . R N D S ( ) | \ { x | y : y x } | / | | . . RR l{l |l \yx{| x |:l yy D(N SD (Sl )( l lN |N (S(\ x|}/ | (21) (21)(21) (21) D SD \l{ . |x }lxx|}}./|| |//|| l l|ll .|| . )D {)) |x : x/ }|:|y:l/:|y l s: (21) R NRDNSR R)|N |l l l|y y| :yy x | (21) }y lare x)\ll |metrics x y[40] yl (21) l |\Sl{(x verify the of HEDA, three performance l l\ { l | adopted (1) performance Overall non-dominated solutions number (ONSN) Denote = 1 2 , ..., L is the union of non-dominated solutions (2) Ratio of non-dominated solutions (RNDS) R N D S ( ) | \ { x l | y : y l l (3)(3) Average distance (DIR) rall non-dominated solutions number (ONSN) Average distance (DIR) (3) Average distance (DIR) the number of non-dominated so Algorithm ,..., L , and thensolutions ONSN ( Denote union1,..., of lnon-dominated with regard to (3) distance (DIR) = (DIR) distance L is the (3) Average distance (DIR) Average distance (3) Average (3) Average distance (DIR) l ) is Average (3)(3) Average distance 1 (DIR) 2 , ...,(DIR) (3) Average distance (DIR) Denote the ratio of solutions which are not dominated by solutions in inated solutions (ONSN) ysolutions ote isdthen the iunion of non-dominated solutions with regard toiny toyin isnormalized the shortest normalized distance from to [40], isand shortest distance solution insolution [40], in =Denote 1 number d2 ,ly )ONSN which are notfrom dominated by as shown inl[40], Equation (20). The lar (..., Denote ) , is the number of non-dominated solutions Algorithm 1,..., ,..., L ( ) Lthe l to y x )( is xDenote i (3) Average distance (DIR) l l to y y is the shortest normalized distance from solution in to d ( π the shortest normalized distance from solution y in [40], y Denote is the shortest normalized distance from solution in [40], Denote is the shortest normalized distance from solution in to ( (xithe yin yx isisthe normalized distance from to [40], y solution Denote normalized distance from solution to [40], l l then Denote normalized distance from solution insolution to [40], i())i )shortest d(i y)xi ()is idthe )dy xdydis yin lthe l d ydx Denote (y xDenote theshortest shortest normalized distance from to l xyyshortest is, better performanc following Equation (21). The higher the value ofin l RNDS ([40], ) l [40], x (i i ) l , DIR and then is in the number of Obviously, non-dominated solutions m l ,..., (defined ) in means is the union of non-dominated solutions with toperformance Obviously, a in better then DIR can be defined in Equation (22).the ) l.means 11,..., , L..., LDIR better of DIR Algorithm aregard smaller abetter then can beONSN defined Equation (22). ONSN ( DIR (smaller are not dominated solutions in as shown in The the value of a abetter lby l means 2which l better l ) is,Obviously, l ) larger can be Equation (22). Obviously, aaEquation smaller a(( convergence a(20). means then DIR can be defined ininin (22). Denote is shortest asmaller smaller aabetter better then DIR can be defined Equation (22). Obviously, smaller means better then DIR can be defined indefined Equation (22). d( ( )( DIR DIR ( ) DIR Obviously, aDIR smaller means anormalized then DIR can be Equation (22). DIR (means Obviously, a smaller ameans better then DIR can be defined in Equation (22). DIR l the Obviously, aObviously, smaller a)l )lla)better then DIR can be defined in Equation (smaller l(22). .Equation Obviously, a means better distance fro then DIR can be defined in Equation (22). ( ) l )iDIR l ) yl x means e notthen dominated solutions in asnon-dominated shown in Equation (20). the valueofof l . convergence performance toasAlgorithm , asdistribution well as aThe better and is the number ofas solutions inoflarger convergence performance to ,well as of well as a better distribution ofdistribution ONSN (lby ) the better performance l . ONSN ( ) lis, l . l . performance to , a better convergence performance to , as well as a better distribution of . . convergence performance to as aabetter better distribution convergence performance toto ,toas well as a,as better distribution of O N SN ( ) of .of | {.lx. l |in y x } | Obviously, convergence performance well as distribution a then DIR be Equation convergence performance , as well awell better of convergence performance to well as a, as better distribution of . : y (22). l .lcan l l \ldefined convergence performance to , as as well asadistribution better distribution l,. as N D S (of l ) l | l l \ {of x ll .| y : y x } | / | l | . better performance Algorithm ted by the solutions in as of shown in Equation (20). The larger the Rvalue l ) is, convergence performance to (20) well ODIR N SN( ( )l ) | \ ({ x l ))/ y x } | ,. y DIR | l d| dy(yx (| |: d(lyx , as(22) . (22)as a better distrib , )/| yl|)/ l )DIR )(2) l | ,.|.,|| DIR ((yx )))/(l dyx ((| )/ || y DIR ( DIR d () )/ |ldyx . . .. (RNDS) (22) ter performance of Algorithm l. DIR ( ( )/ | y,.,y DIR ( ) )/ | | , (22) (22) ( ) d | ld l, y y (22) , . (22) yx l l Ratio of non-dominated solutions (22) DIR d )/ | l l y l yx (22) l yx l . y O N SN ( l ) | l \ { x l | y (3) : Average y x } | yxdistance l (20) (22) (DIR) DIR ( ) d (l )/ | | , y l yxby solutions . Denote the ratio(20) of solutions which are not dominated solutions in O N SN ((2) l )Ratio | l of \ { xnon-dominated l | y : y x } |(RNDS) y in Denote is the shortest normalized distance from solution 5.3. Comparisons Results 5.3. Comparisons Results d y x ( i ) 5.3. Comparisons Results o of non-dominated solutions (RNDS) 5.3. Comparisons Results 5.3. Comparisons Results 5.3. Comparisons Results 5.3. Comparisons Results 5.3. Comparisons Results 5.3. Comparisons is, the better following Equation (21). The higher value of RNDS ( ) Denote 5.3. theResults ratio of solutions not dominated by solutions in the , shown as Comparisons Results which are l (22). Obviously, a smaller DIR(performa then DIR can be defined in Equation l ) m minated solutions (RNDS) ote the5.3.1. ratio of solutions which are not dominated by solutions in , shown as 5.3. Comparisons Results l.Algorithms 5.3.1. Comparison Existing Algorithms 5.3.1. Comparison with Existing better performance of Algorithm following Equation (21). The with higher the value of RNDS (l ) is, the Comparison with Existing Algorithms 5.3.1. Comparison with Existing Algorithms 5.3.1. Comparison with Existing Algorithms 5.3.1. Comparison with Existing Algorithms convergence performance to , as well as a better distribution of l . 5.3.1. Comparison with Existing Algorithms 5.3.1. Comparison with Existing Algorithms 5.3.1. Comparison with Existing Algorithms 5.3.1. Comparison Existing Algorithms the better performance of Algorithm gof Equation The higher the value ofwith RNDS (l ) is, solutions which are not dominated bycompare solutions in HEDA , shown as(algorithms: three l. (21). with In In this subsection, we HEDA algorithms: R N5.3.1. D S(1) Comparison HEDA_noLS, | HEDA_noLS, l \(1) { xwith HEDA_noLS, yvariant aofvariant :HEDA y x } of | / | l | . this subsection, we compare with three In this subsection, we compare HEDA with three algorithms: a variant ofAlgorithms l ) (1) l |a Existing In this subsection, we compare HEDA with three algorithms: (1) HEDA_noLS, aaof variant ofofof In this subsection, we compare HEDA with three algorithms: (1) HEDA_noLS, aavariant variant In this subsection, we compare HEDA with three algorithms: (1) HEDA_noLS, avariant variant In subsection, we compare HEDA with three algorithms: (1) HEDA_noLS, In this subsection, we compare HEDA with three algorithms: (1) HEDA_noLS, a variant of Inthe this we compare HEDA with three algorithms: (1) HEDA_noLS, aparameter of Inthis this we compare HEDA with three algorithms: (1) HEDA_noLS, variant of is, the better performance of Algorithm 1). The HEDA higher value of RNDS (subsection, )the DIR ( ) d ( )/ | | insubsection, which the PIN-based local search operator is removed, and the parameter setting is the same as HEDA in which PIN-based local search operator is removed, and the setting is the , . in which the PIN-based local search operator is removed, and the parameter setting is the l y yx parameter l Rin N PIN-based D S ( the the ) PIN-based | PIN-based l \ { search x local local local | search y operator operator : y xis } | is / |and l the | .l parameter (21) l the l operator HEDA in which removed, and the setting is the HEDA which search removed, and the parameter setting is the HEDA which the local is removed, setting is the HEDA in which PIN-based search operator is removed, and the parameter setting is the HEDA in which the PIN-based local search operator is removed, and the parameter setting is the HEDA inin which the PIN-based local search operator is removed, and the parameter setting is the In this subsection, we compare HEDA HEDA in which[41], the local search operator is removed, and the parameter setting is thewith three algo HEDA; (2) NSGAII aPIN-based non-dominated sorting multi-objective genetic algorithm, which (2) NSGAII a| non-dominated non-dominated sorting multi-objective genetic sameR NasD SHEDA; (2) [41], awell-known sorting multi-objective genetic (DIR) . ( same |as as as \ {HEDA; x | NSGAII yNSGAII well-known a[41], : [41], y(3) aa x }Average /well-known | non-dominated lnon-dominated | distance (21) well-known same HEDA; (2) well-known non-dominated sorting multi-objective genetic l )HEDA; l HEDA; l [41], same as HEDA; (2) NSGAII [41], aawell-known well-known non-dominated sorting multi-objective genetic same HEDA; (2) NSGAII [41], awell-known sorting multi-objective genetic same (2) [41], non-dominated sorting multi-objective genetic same as (2) NSGAII [41], well-known sorting multi-objective genetic same asasHEDA; (2) NSGAII a non-dominated sorting multi-objective genetic HEDA in which the PIN-based local search same as HEDA; (2) NSGAII [41], well-known non-dominated sorting multi-objective genetic hasalgorithm, been widely used forused various multi-objective optimization problems;optimization and (3) PGA which has been widely usedmulti-objective for various multi-objective problems; and (3)operator is remo algorithm, which has been widely for various optimization problems; and[31], (3) a relatively (3) | algorithm, Average (DIR) ynon-domi has used for various multi-objective optimization problems; and (3) .for Denote isoptimization the shortest normalized distance from solution in algorithm, which been widely used for various multi-objective optimization problems; and (3) been widely for various multi-objective optimization problems; and (3) (3) D S ( algorithm, l \ { xalgorithm, which distance which yhas which been widely :has yhas been xused } |used /widely |widely for 5.3. dResults ) (21) algorithm, which been used for various optimization problems; and (3) has widely used various multi-objective optimization problems; and which been multi-objective problems; and (3) Comparisons l ) algorithm, l |has l |various same as HEDA; (2) NSGAII [41], a well-known y x ( i multi-objective algorithm, which has been widely used for various multi-objective optimization problems; and (3) reported algorithm for a special kind of intermodal transportation problem which is quite PGA [31], agenetic relatively new reported genetic algorithm for aofspecial kind of intermodal transportation PGA [31], anew relatively new reported genetic algorithm for a special kind intermodal transportation rage distance (DIR) PGA [31], a relatively new reported genetic algorithm for a special kind of intermodal transportation PGA [31], a relatively new reported genetic algorithm for a special kind of intermodal transportation PGA [31], a relatively new reported genetic algorithm for a special kind of intermodal transportation PGA [31], a relatively new reported genetic algorithm for a special kind of intermodal transportation PGA [31], a relatively new reported genetic algorithm for a special kind of intermodal transportation PGA [31], a relatively new reported genetic algorithm for a special kind of intermodal transportation y algorithm, which has been used for various multi-ob Denote is shortest normalized distance from solution in to [40], PGA [31], a the relatively new reported algorithm forthe a special of intermodal transportation Obviously, a the smaller then DIR can in Equation (22). DIR ( l ) are d is to l widely the similar It should begenetic mentioned that all algorithms executed based on y x (quite i )MO_MSITNDP. problem which is quite similar to MO_MSITNDP. Itdefined should be mentioned that all the algorithms are problem which similar to MO_MSITNDP. It should bebe mentioned that allkind algorithms are problem which isissimilar quite similar to MO_MSITNDP. Itbe should be mentioned that all the algorithms are problem which is quite similar MO_MSITNDP. It should be mentioned that all the algorithms are problem which is quite similar to MO_MSITNDP. It should mentioned that all the algorithms are 5.3.1. Comparison with Existing Algorithms problem which quite similar toto MO_MSITNDP. It should be mentioned that all the algorithms are (DIR) problem which is quite to MO_MSITNDP. It should be mentioned that all the algorithms are problem which is quite similar to MO_MSITNDP. It should be mentioned that all the algorithms are y PGA [31], a relatively new reported genetic algorithm for problem which is quite similar to MO_MSITNDP. It should be mentioned that all the algorithms are ote is the shortest normalized distance from solution in to [40], d yexecuted ( ) convergence performance well as a better modified model inon Section 3.2. l ,)as means xthen i executed based the modified model Obviously, a 3.2. smallerto DIR betterdistribution of l . a sp DIR can bethe defined in Equation (22). based on modified model in Section 3.2.in Section ( l executed based on the modified model in Section 3.2. executed based on the modified model in Section 3.2. executed based on the modified model in Section 3.2. executed based on the modified model in Section 3.2. executed based on the modified model in Section 3.2. executed based on the modified model in Section 3.2. problem which is quite similar tomeanwhile, MO_MSITNDP. It should be executed based on the modified model insubsection, Section yaas For the parameters setting, we set the population size PS = 50 all algorithms, and, In this we compare HEDA three algorithms: the shortest distance from solution in to size [40], PS algorithms, 50 with 50 Obviously, smaller means a.for better Ris can be defined Equation (22). For the parameters setting, set the size for all algorithms, and,(1) HEDA_noLS DIR (population 3.2. For normalized the in parameters setting, the population for all and, convergence performance to , setting, asset well awe better distribution of ll )PS we l 50 PS 50 For the parameters we set the population size for all algorithms, and, PS 50 PS 50 For the parameters setting, we set the population size for all algorithms, and, For the parameters setting, we set the population size for all algorithms, and, PS 50 PS For the parameters setting, we set the population size for all algorithms, and, For the For parameters setting, we set population the population size for all and, PS DIR 50 For the parameters setting, we set the size for algorithms, and, executed based modified model in Section PS on 50 ( dofthe (algorithms, | , are theHEDA’s parameters setting, we set the population size for all|and algorithms, and, y .the3.2. HEDA’s learning LR =0.06, the crossover and mutation probability of NSGAII PGA set HEDA in) which the PIN-based local search operator is removed, and parameter l )all yx l )/ LR 0.06 LR 0.06 meanwhile, learning rate , lthe and mutation probability of NSGAII and meanwhile, HEDA’s rate , the crossover and mutation probability NSGAII and nce performance to ,learning asHEDA’s well rate as alearning better distribution of .crossover Obviously, smaller means acrossover better fined in Equation (22). DIR (the l 0.06 LR meanwhile, rate , the and mutation probability of NSGAII and LR 0.06 LR 0.06 meanwhile, HEDA’s learning rate , the crossover and mutation probability of NSGAII and meanwhile, HEDA’s learning rate , crossover and mutation probability of NSGAII and LR 0.06 LR 0.06 meanwhile, HEDA’s learning rate , the crossover and mutation probability of NSGAII and meanwhile, HEDA’s learning rate , the crossover and mutation probability of NSGAII and LR 0.06 meanwhile, HEDA’s learning rate , the crossover and mutation probability of NSGAII and For the parameters setting, we set the population si LR 0.06 meanwhile, HEDA’s learning rate , the crossover and mutation probability of NSGAII and to 0.8 and 0.2, respectively. It should be noted that we have embedded the PIN-based local search same as HEDA; (2) NSGAII [41], a well-known non-dominated sorting multi-obj DIR ( ) d ( )/ | | , that . PGA are set 0.8 and 0.2, respectively. It should be noted that we have (22)the PIN-based are set to 0.8 and 0.2,to respectively. It should be noted we have embedded theembedded PIN-based y l 0.2, yx l nce to PGA , as well as a better distribution of . PGA are set to 0.8 and respectively. It should be noted that we have embedded the PIN-based PGA are set to 0.8 and 0.2, respectively. It should be noted that we have embedded the PIN-based PGA are set to 0.8 and 0.2, respectively. It should be noted that we have embedded the PIN-based l PGA are set to 0.8 and 0.2, respectively. It should be noted that we have embedded the PIN-based are set 0.8to 0.2, respectively. Ittoshould noted that we have embedded the PIN-based PGAPGA are set to 0.8to and 0.2, should be noted that we have embedded PIN-based 0.06optimization meanwhile, HEDA’s learning rate , the crossover an are to 0.8 and 0.2, Itbe should be noted that we embedded theLR PIN-based operator the NSGAII and PGA ensure the of comparisons. has been widely used for various multi-objective prob DIR (PGA in dand (respectively. |Itrespectively. local operator to the NSGAII and PGA to ensure the fairness ofthe comparisons. , algorithm, .which (22)have local search operator inset to the NSGAII and PGA to ensure thefairness fairness of comparisons. y5.3. l )search yx l )/in| in Comparisons Results local search operator to the NSGAII and PGA tothe ensure the fairness ofof comparisons. local search operator in to the NSGAII and PGA to ensure the fairness of comparisons. local search operator in to the NSGAII and PGA to ensure fairness of comparisons. local search operator in to the NSGAII and PGA to ensure the fairness comparisons. local search operator in to the NSGAII and PGA to ensure the fairness of comparisons. local search operator in to the NSGAII and PGA to ensure the fairness of comparisons. PGA are set to 0.8 and 0.2, respectively. It should be noted local search operator in to the NSGAII and PGA to ensure the fairness of comparisons. WeWe first make observation on[31], thethe approximate Pareto fronts with regard to to thefor first instance PGA a relatively new reported genetic algorithm afirst special kind of intermodalth t first make an observation on approximate Pareto fronts regard the instance first make an observation on the. approximate Pareto fronts with regard towith the first instance DIR(l We ) We dfirst ( |first | , an (22) yan an yxmake l )/ We first make an observation on the approximate Pareto fronts with regard to the first instance We first make observation on the approximate Pareto fronts with regard to the first instance make an observation on the approximate Pareto fronts with regard to the first instance We make observation on the approximate Pareto fronts with regard to the first instance We first make an observation on the approximate Pareto fronts with regard to the first instance 5.3. Comparisons Results We first an observation on the approximate Pareto fronts with regard to the first instance local search operator in to the NSGAII and PGA to ensure the We first make an observation on the approximate Pareto fronts with regard to the first instance of 10 × 5, 20 × 5, 30 × 7, 40 × 9 and 50 × 13, espectively. Furthermore, the obtained results of HEDA, problem which is quite similar to MO_MSITNDP. It should be mentioned that all the a 5 20 40 5 7 40 40 9, espectively. 50 13 , Furthermore, 510 30 95.3.1. 13 , 720 , 930and , 50 and espectively. Furthermore, obtained results of of 10 5 , 20of10 the obtained the results of Comparison with Existing Algorithms ,,55 , 30 97and can 50 50 of ,5 ,7 , ,espectively. Furthermore, the obtained ofofof 20 55, 40 30 797, ,,40 40 99and 50 Figure 13 10 20 50 13 , , and ,espectively. espectively. Furthermore, the obtained results of10 and ,13 espectively. Furthermore, the results 5,30 5,40 7,PGA 13 950 13 13 10,5HEDA_noLS, ,of 5of 20 30 40 espectively. , 30 Furthermore, the obtained results arisonsof Results of ,10 and , 50 espectively. Furthermore, the obtained results of 5 20 510 7NSGAII 9and 50 ,75 ,and the obtained results ofofresults We first make an observation on the approximate Pareto 10 5NSGAII 20 30 40 13 of ,20 ,30 and ,9.Furthermore, espectively. Furthermore, the obtained results of be seen in 9. based on the modified model inobtained Section 3.2. HEDA, HEDA_noLS, NSGAII and PGA be seen in Figure 9. HEDA, HEDA_noLS, and PGA canexecuted be seen incan Figure HEDA, HEDA_noLS, NSGAII and PGA can be seen in Figure 9. 5.3.1. Comparison with Existing Algorithms HEDA, HEDA_noLS, NSGAII and PGA can be seen in Figure 9. HEDA, HEDA_noLS, NSGAII and PGA can be seen in Figure 9. HEDA, HEDA_noLS, NSGAII and PGA can be seen in Figure 9. HEDA, HEDA_noLS, NSGAII and PGA can be seen in Figure 9. In this subsection, we compare HEDA with three algorithms: (1) HEDA_no HEDA, HEDA_noLS, NSGAII and PGA can be seen in Figure 9. 10set in59.obtained 20 5set 30HEDA 7 , 40 9 size 50are 1350, espectively. ofFigure ,we ,the and are HEDA, NSGAII and PGA bethe seen in Itobviously is be be obviously seen Figure 9can that Pareto obtained byby HEDA PS s For setting, set population for all alg It can isHEDA_noLS, can obviously seen in Figure 9parameters that the solutions in Pareto set obtained HEDA It is can be seen in Figure 9in that thethe solutions insolutions Pareto by are ItAlgorithms is can be obviously seen inin Figure 9the that the solutions in Pareto set obtained by HEDA are mparison with It is can be obviously seen in Figure 99that that the solutions in Pareto set obtained by HEDA are is can be obviously seen in Figure 9HEDA that the solutions in Pareto set obtained by HEDA are It is can be obviously seen Figure 9 the solutions in Pareto set obtained by HEDA are It is can be obviously seen in Figure 9 that solutions in Pareto set obtained by HEDA are ItInIt isExisting can be obviously seen in Figure 9 that the solutions in Pareto set obtained by HEDA are in which the PIN-based local search operator is removed, and the HEDA, HEDA_noLS, NSGAII and PGA can be seen inparame Figureo It is can be obviously seen in Figure that the solutions in Pareto set obtained by HEDA are better (both in terms of the quality and quantity of solutions) than the ones from the three other LR 0.06 this in subsection, we compare HEDA withand algorithms: (1)rate HEDA_noLS, athree variant of and meanwhile, HEDA’s learning , the crossover better in quality terms ofand the quality quantity of solutions) than the from the threemutation other probability better (both terms(both of the quantity ofthree solutions) than the ones from theones other better (both in terms of the quality and quantity of solutions) than the ones from the three other better (both in terms of the quality and quantity of solutions) than the ones from the three other better (both in terms of the quality and quantity of solutions) than the ones from the three other better (both in terms of the quality and quantity of solutions) than the ones from the three other better (both in terms of the quality and quantity of solutions) than the ones from the three other better (both in terms of the quality and quantity of solutions) than the ones from the three other same as HEDA; (2) NSGAII [41], a well-known non-dominated sorting multi-o It is can be obviously seen in Figure 9 that the solution h Existing Algorithms better inallcases terms ofofvariable the quality and quantity of solutions) than ones the three other in all of problem which implies HEDA share aafrom high superiority HEDA in which the(both PIN-based local search operator is removed, and parameter setting isnoted the PGA are set(1) tosizes, 0.8 and 0.2, respectively. It should be that we have his subsection, wealgorithms HEDA with three algorithms: HEDA_noLS, athe variant ofthe algorithms cases variable problem sizes, which implies HEDA share high superiority in embedded t algorithms incompare all cases ofin variable problem sizes, which implies HEDA share a high superiority in algorithms inof all cases of variable problem sizes, which implies HEDA share aasuperiority high superiority ininin algorithms cases variable problem sizes, which implies HEDA share aahigh high superiority algorithms in all cases of variable problem sizes, which implies HEDA share afour high superiority in listed algorithms inin allall cases ofof variable problem sizes, which implies HEDA superiority algorithms in all cases of variable problem sizes, which implies HEDA share ashare high in algorithms in all cases variable problem sizes, which implies HEDA share a high superiority in algorithm, which has been widely used for various multi-objective optimization p better (both in terms of the quality and quantity of solution algorithms in all cases of variable problem sizes, which implies HEDA share high superiority inof in solving MO_MSITNDP. Furthermore, the statistic results of these algorithms are in same as HEDA; (2) NSGAII [41], a well-known non-dominated sorting multi-objective genetic local search operator in to the NSGAII and PGA to ensure the fairness comparisons. n which the PIN-based local search operator is removed, and the parameter setting is the solving MO_MSITNDP. Furthermore, the statistic results of these four are listed in Tables solving MO_MSITNDP. Furthermore, the(1) statistic results ofathese fourofalgorithms arealgorithms listed in Tables n, we compare HEDA with three algorithms: HEDA_noLS, variant solving MO_MSITNDP. Furthermore, the statistic results ofofalgorithms these four algorithms are listed inin Tables solving MO_MSITNDP. Furthermore, the statistic results of these four algorithms are listed in Tables solving MO_MSITNDP. Furthermore, the statistic results of these four algorithms are listed in Tables solving MO_MSITNDP. Furthermore, the statistic results these four algorithms are listed Tables solving MO_MSITNDP. Furthermore, the statistic results of these four algorithms are listed in Tables solving MO_MSITNDP. Furthermore, the statistic results of these four are listed in Tables PGA [31], a relatively new reported genetic algorithm for a special kind of intermoda algorithms in all cases of variable problem sizes, which impl solving MO_MSITNDP. Furthermore, the statistic results of these four algorithms are listed in Tables Tables 1–4. algorithm, which has widely used multi-objective optimization and Pareto (3) We first make an observation on theproblems; approximate fronts with regard to the HEDA; (2) NSGAII [41], a been well-known non-dominated sorting multi-objective genetic 1–4. 1–4. IN-based local search operator is removed, andfor thevarious parameter setting is the 1–4. 1–4. 1–4. 1–4. 1–4. problem which is quite similar toobviously MO_MSITNDP. It should be mentioned thatobtain all 1–4. solving MO_MSITNDP. Furthermore, the statistic results of th 1–4. We used cannew see invarious Tablesgenetic 1–3 that the 5 performance ofproblems; HEDA is better than otherFurthermore, three PGA [31], a relatively reported algorithm for a special kind of intermodal transportation m, which has been widely for multi-objective optimization and (3) 10 20 5 30 7 40 9 50 13 of , , , and , espectively. the NSGAII [41], a well-known non-dominated sorting multi-objective genetic executed based on mentioned the modified model inalgorithms Section 3.2. 1–4. compared algorithm with ONSN, RNDS and DIR metrics. It clear that HEDA’s problem which is quite similar toassociated MO_MSITNDP. It problems; should be that all the are , a relatively new reported genetic algorithm for aHEDA, special kind of intermodal HEDA_noLS, NSGAII and PGA can beisseen in Figure 9. been widely used for various multi-objective optimization and (3)transportation PS set 50 obtained For the parameters setting, we set the population size for all a performance is more superior than its variant HEDA_noLS, which demonstrate effectiveness of executed based on the modified model in Section 3.2. which is quite similar to MO_MSITNDP. It should be mentioned that all the algorithms are It is can be obviously seen in Figure 9 that the solutions inour Pareto new reported genetic algorithm for a special kind of intermodal transportation LR 0.06 meanwhile, HEDA’s , theofcrossover and mutation probability proposed PIN-based local search. For most instances, canrate find more non-dominated PS 50 ontothe modified model in Section 3.2. For the parameters setting, we setbetter the sizeHEDA for and all algorithms, and, solutions (both terms oflearning the quantity solutions) than the ones from th ebased similar MO_MSITNDP. It should be mentioned thatpopulation all theinalgorithms are quality PGA are set to 0.8 and 0.2, respectively. It should be noted that we have embedde PS 50 LR 0.06 the parameters setting, we set the population size for all algorithms, and, algorithms in all cases of variable problem sizes, which implies HEDA share a high meanwhile, HEDA’s learning rate , the crossover and mutation probability of NSGAII and modified model in Section 3.2. local search operator in to the NSGAII and PGA to ensure the fairness of compariso solving MO_MSITNDP. Furthermore, the statistic results of these four algorithms are l LR 0.06 PGA are set to 0.8 and 0.2, respectively. It should be noted that we have embedded the PIN-based ile, HEDA’s rate population, the probability of NSGAII and PS 50and ers setting, learning we set the sizecrossover formutation all algorithms, and, We first make an observation on the approximate Pareto fronts with regard to local search operator in to NSGAII and1–4. PGA the fairness of comparisons. set to 0.8 and respectively. It the should be noted that to weensure have embedded PIN-based LR0.2, 0.06 earning rate , the crossover and mutation probability of NSGAII andthe 10 5 20 5 30 7 40 9 50 13 , Pareto , , regard and , espectively. Furthermore, the obt We anbeand observation on approximate fronts with to the first instance operator in tofirst the NSGAII PGAthat to ensure theof fairness of comparisons. dch 0.2, respectively. It make should noted wethe have embedded the PIN-based HEDA, HEDA_noLS, NSGAII and PGA can be seen 10observation 5 , 20 5 ,on30 7approximate 5013 first make the Pareto fronts with regard to the first instance of an , 40 9 and , espectively. Furthermore, the obtained resultsinofFigure 9.
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We can see in Tables 1–3 that the performance of HEDA is obviously better than other three compared algorithm associated with ONSN, RNDS and DIR metrics. It is clear that HEDA’s Sustainability 2017, 9, 1133 15 of 24 performance is more superior than its variant HEDA_noLS, which demonstrate effectiveness of our proposed PIN-based local search. For most instances, HEDA can find more non-dominated solutions than thanthe theother otheralgorithms, algorithms,and anditsitssolutions solutionspossess possessa ashorter shorterdistance distancetotothe theunion unionofofnon-dominated non-dominated solutions. HEDA ensure anan outstanding convergence toto the optimal Pareto font, solutions.Thus, Thus,not notonly onlydoes does HEDA ensure outstanding convergence the optimal Pareto font, but it also keeps a reasonable distribution and diversity on its approximated Pareto font. The feasible but it also keeps a reasonable distribution and diversity on its approximated Pareto font. The feasible ratio ratioofofHEDA HEDAreaches reachestoto100%, 100%,asaswell wellasasother otheralgorithms, algorithms,which whichdemonstrated demonstratedthe thereliability reliabilityofofour our proposed Accordingly,the the proposed HEDA canseen be as seen as an effective solution for proposed HEDA. HEDA. Accordingly, proposed HEDA can be an effective solution for addressing addressing MO_MSITNDP. MO_MSITNDP. InInterms HEDA has lower runtimes than NSGAII and PGA, termsofofthe theruntime, runtime,Figure Figure1010shows showsthat that HEDA has lower runtimes than NSGAII and PGA, which indicates that the running speed of HEDA is faster than NSGAII and PGA. The runtimes which indicates that the running speed of HEDA is faster than NSGAII and PGA. The runtimesofof HEDA HEDAare area alittle littlebit bithigher higherthan thanHEDA_noLS, HEDA_noLS,implying implyingthat thatthe thePIN-based PIN-basedlocal localsearch searchofofHEDA HEDA causes causesrelative relativelower lowertime timeconsumption. consumption.InInaddition, addition,the thefeasibility feasibilityrates ratesfor forall allalgorithms algorithmsare are100%. 100%.
(a)
(b)
(c)
(d)
(e)
30 7 5× 2020 5×; (c) Figure The Obtained approximate Pareto fonts of all (a) 10 ; (b) Figure9. 9. The Obtained approximate Pareto fonts ofalgorithm. all algorithm. (a)10 5; (b) 5; (c) 30 × 7; 40×9;9 ;(e)(e)5050 13. ; (d) (d) 40 × 13.
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Table 1. Comparisons of HEDA, NSGAII, PGA and HEDA-noLS (ONSN). HEDA vs. NSGAII NSGAII
Instance 5×3 10 × 3 10 × 4 10 × 5 20 × 4 20 × 5 20 × 6 20 × 7 30 × 6 30 × 7 30 × 8 30 × 9 40 × 9 40 × 11 50 × 13 Average
HEDA vs. PGA HEDA
PGA
HEDA vs. HEDA_noLS HEDA
HEDA_noLS
HEDA
MAX
AVG
SD
MAX
AVG
SD
MAX
AVG
SD
MAX
AVG
SD
MAX
AVG
SD
MAX
AVG
SD
44 53 52 36 0 0 0 0 0 0 0 0 0 0 0 12.33
20.24 6.29 3.76 6.81 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 2.47
14.45 13.99 12.22 12.55 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 3.55
5 8 10 9 9 8 8 10 5 7 4 4 12 6 9 7.60
3.71 5.48 5.81 4.86 3.81 3.67 4.38 4.48 2.52 2.38 1.95 2.00 5.33 2.33 2.00 3.65
0.63 1.65 1.87 2.05 1.94 1.98 1.76 1.99 1.18 1.40 1.09 1.07 3.48 1.32 1.69 1.67
3 2 0 1 0 0 0 0 0 0 0 0 0 0 0 0.40
1.57 0.10 0.00 0.14 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.12
0.79 0.43 0.00 0.35 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.10
5 6 12 9 9 9 8 8 7 5 5 6 11 6 6 7.47
3.81 4.57 6.52 4.62 4.24 3.86 3.76 3.48 2.48 2.48 2.33 2.10 4.00 3.14 2.05 3.56
0.59 1.22 2.34 2.08 1.82 1.61 1.69 1.94 1.40 1.33 1.28 1.38 2.98 1.39 1.13 1.65
4 8 9 6 4 2 6 3 6 3 3 3 2 1 0 4.00
2.52 2.76 2.43 2.00 0.71 0.52 1.19 0.29 0.86 0.67 0.29 0.29 0.10 0.10 0.00 0.98
1.26 2.04 2.57 1.90 1.16 0.73 1.82 0.70 1.83 0.99 0.76 0.76 0.43 0.29 0.00 1.15
4 8 9 6 7 7 8 7 7 4 8 3 10 5 10 6.87
2.81 3.81 5.67 3.43 4.00 3.95 3.71 3.76 2.48 2.00 2.29 1.67 3.52 1.86 2.48 3.16
0.96 2.38 2.40 1.47 1.57 1.43 1.91 1.74 1.65 1.07 1.69 0.64 3.19 1.04 2.15 1.69
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Table 2. Comparisons of HEDA, NSGAII, PGA and HEDA-noLS (RNDS). HEDA vs. NSGAII NSGAII
Instance 5×3 10 × 3 10 × 4 10 × 5 20 × 4 20 × 5 20 × 6 20 × 7 30 × 6 30 × 7 30 × 8 30 × 9 40 × 9 40 × 11 50 × 13 Average
HEDA vs. PGA HEDA
PGA
HEDA vs. HEDA_noLS HEDA
HEDA_noLS
HEDA
MAX
AVG
SD
MAX
AVG
SD
MAX
AVG
SD
MAX
AVG
SD
MAX
AVG
SD
MAX
AVG
SD
0.44 0.53 0.54 0.36 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.12
0.21 0.07 0.04 0.07 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.03
0.15 0.14 0.13 0.13 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.04
1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
1.00 1.00 0.99 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
0.00 0.00 0.02 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.67 0.25 0.00 0.25 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.08
0.33 0.01 0.00 0.04 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.03
0.18 0.05 0.00 0.09 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.02
1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
1.00 1.00 1.00 1.00 0.75 0.50 1.00 0.50 0.86 1.00 0.75 1.00 1.00 1.00 0.00 0.82
0.68 0.55 0.37 0.43 0.17 0.13 0.20 0.06 0.13 0.17 0.07 0.12 0.05 0.07 0.00 0.21
0.35 0.38 0.33 0.38 0.26 0.19 0.30 0.13 0.27 0.26 0.19 0.30 0.21 0.23 0.00 0.25
1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
0.78 0.69 0.85 0.99 1.00 0.99 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.95
0.24 0.34 0.30 0.05 0.00 0.03 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.06
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Table 3. Table 3. Comparisons of HEDA, NSGAII, PGA and HEDA-noLS (DIR). HEDA vs. NSGAII NSGAII
Instance 5×3 10 × 3 10 × 4 10 × 5 20 × 4 20 × 5 20 × 6 20 × 7 30 × 6 30 × 7 30 × 8 30 × 9 40 × 9 40 × 11 50 × 13 Average
HEDA vs. PGA HEDA
PGA
HEDA vs. HEDA_noLS HEDA
HEDA_noLS
MIN
AVG
SD
MIN
AVG
SD
MIN
AVG
SD
MIN
AVG
SD
36.64 181.89 269.57 1598.90 1054.05 1216.20 1650.11 1343.27 2254.08 1785.60 3116.28 3917.91 5795.40 2128.72 4697.53 2069.74
123.36 544.88 1225.25 3824.40 2843.73 4652.32 5755.40 8610.60 6060.90 6760.44 8257.92 9219.88 19,982.45 7800.28 10,692.39 6423.61
67.77 265.34 758.57 1870.46 1558.43 3060.08 2848.66 3947.62 3079.91 4068.56 4925.34 5022.42 10,818.12 4562.72 6753.19 3573.81
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
1107.72 2286.86 1550.08 9359.46 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 953.61
1012.93 5945.60 6811.30 17,387.86 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 2077.18
61.66 158.87 647.41 2038.44 1954.42 2946.40 1967.32 2449.42 2639.69 4635.60 5636.08 6149.95 9074.13 6379.80 10,426.97 3811.08
116.48 793.87 1960.32 4826.59 4134.30 6184.59 7092.64 10,036.65 8937.82 11,795.78 15,186.47 15,340.45 34,600.99 20,857.98 22,517.95 10,958.86
74.67 400.31 800.04 2376.99 1889.59 2407.24 3234.82 6093.77 5210.97 6546.49 8625.25 10,717.45 21,887.46 9415.57 12,695.50 6158.41
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
60.73 51.70 55.16 246.67 0.00 0.00 203.33 504.80 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 21.28 53.54
HEDA
MIN
AVG
SD
MIN
AVG
SD
0.00 0.00 0.00 32.09 157.67 1096.54 217.86 833.73 1347.68 1730.08 2451.58 2664.78 4809.12 3564.77 5898.19 1653.61
15.66 183.72 390.81 829.86 1649.53 2973.23 2257.73 4881.32 4364.83 4839.59 8163.42 5933.61 15,213.67 7778.40 16,269.53 5049.66
19.03 191.64 331.60 593.75 1260.86 1518.30 1475.94 2931.44 2613.23 3264.24 6419.59 2339.85 10,688.43 5030.00 11,700.77 3358.58
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
25.87 128.18 93.72 652.71 233.46 404.02 691.68 371.87 1512.26 1572.35 1162.14 1003.98 525.18 439.48 0.00 587.79
53.26 163.28 138.05 968.25 401.39 581.21 1160.49 954.23 3231.31 2319.77 3194.25 2603.61 2348.70 1358.51 0.00 1298.42
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Figure 10. 10. The Thetrends trendsofof run time algorithms. The feasibility these four algorithms Figure run time for for the the algorithms. * The*feasibility rates ofrates theseof four algorithms are 100%. are 100%.
5.3.2. Comparison with Optimization Solver 5.3.2. Comparison with Optimization Solver In this subsection, the developed HEDA is compared with optimization solver. Owing to the this subsection, themathematical developed HEDA is compared optimization Owing solver to the linearIncharacteristic of the model, we choose with CPLEX 12.5, whichsolver. is a popular linear characteristic of the mathematical model, we choose CPLEX 12.5, which is a popular solver used in optimization area. The multiple objectives are transformed into a single objective problem used in optimization area. The multiple are transformed into a single by using the epsilon constraint method objectives [42] to satisfy the solving conditions of objective CPLEX. Itproblem should by be using the epsilon constraint method [42] to satisfy the solving conditions of CPLEX. It should be noted that we do not purse the optimal solutions regarding the branch-and-cut phase addressed by noted that we do more not purse the solutions optimal solutions regarding the branch-and-cut phase addressed by CPLEX to obtain feasible to construct the approximate Pareto font. That is, once the CPLEX to obtain more feasible solutions to construct the approximate Pareto font. That is, once the feasible solution is found, CPLEX will stop and turn to solve the next problem in terms of another feasiblebound. solution is found, CPLEX will stop and turn to solve theofnext problem inDIR terms ofTime another upper Upper limit of time for CPLEX is 600 s. The results ONSN, RNDS, and are upper bound. Upper limit of time for CPLEX is 600 s. The results of ONSN , RNDS , DIR and Time are shown in Table 4. shown in Table 4. In Table 4, we can find that, for the small scale instances, HEDA has lower runtimes and the quality Table 4, iswe canatfind that, for the small instances, HEDA hasHEDA lower has runtimes and the of theInsolutions also an acceptable level. Forscale the middle problem size, advantages in quality ofand thecan solutions also at an acceptable level. Furthermore, For the middle problem size, to HEDA runtimes provideisbetter solutions than CPLEX. CPLEX is unable solve has the advantages in runtimes and provide better solutions CPLEX. Furthermore, is unable large scale instances, such ascan instance 40 × 11 and 50 × 13,than whereas HEDA can obtainCPLEX solutions easily, 40 11 50 13 to solve the large scale instances, such as instance and , whereas HEDA can obtain which indicates HEDA is more capable with respect to all instances. solutions easily, which indicates HEDA is more capable with respect to all instances. Table 4. Comparisons of HEDA and CPLEX under a single run. Table 4. Comparisons of HEDA and CPLEX under a single run. CPLEX CPLEX Instance Instance ONSNONSN RNDS RNDS DIRDIR 5×3 10 × 3 10 × 4 10 × 5 20 × 4 20 × 5 20 × 6 20 × 7 30 × 6 30 × 7 30 × 8 30 × 9 50 × 13 40 × 11 50 × 13
5 3 47 * 10355 * 61 * 10438 * 105 8 204 11 205 9 7 206 4 207 7 306 3 4 307 5 308 309 50 13 40 11 50 13
47 * 1.00 55 * 1.00 1.00 61 * 1.00 38 * 0.88 8 0.87 0.74 11 0.89 9 0.77 7 0.74 4 0.52 0.69 7 0.56 3 4
1.00 0.000.00 1.00 0.000.00 0.00 1.00 0.00 0.00 1.00 879.01 0.00 0.88 669.25 879.01 420.21 0.87 669.25 321.52 0.74 264.67 420.21 0.89 74.45 321.52 254.91 0.77 264.67 141.12 0.74 957.76 74.45 0.52 -254.91 -141.12 0.69
Time Time
ONSN ONSN
