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Freight transportation network model with bundling option
Transportation Research Part E 133 (2020) 101827
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Transportation Research Part E journal homepage: www.elsevier.com/locate/tre
Freight transportation network model with bundling option Young-Soo Myung, Yung-Mok Yu
⁎,1
T
Department of Business Administration, Dankook University, Yongin-si, Gyeonggi-do 16890, Republic of Korea
ARTICLE INFO
ABSTRACT
Keywords: Network flow Bundling option Relocating foldable containers
We consider a freight transportation network model in which we can send goods with or without bundling. When we choose the bundling option, we can send multiple goods, after grouping into a bundle, at the cost of one unit good; however, bundling and unbundling also incur costs. The model was originally developed to solve the problem of relocating empty containers in hinterland transportation, but it also has an application in multimodal freight transportation. We investigate the complexity of the problem and develop an efficient heuristic that uses a network flow algorithm.
1. Introduction In this paper, we consider the Freight Transportation Network Problem with the Bundling option (FTNPB), which was proposed by Shintani et al. (2010) to address the problem of relocating empty containers in hinterland transportation. The FTNPB is a variant of the single commodity network flow problem with multiple sources and sinks. The problem involves sending flows from a set of source nodes to satisfy the demands at the sink nodes at minimum cost, but unlike in a standard problem, we can group a certain number of flows into a bundle and send each bundle at the cost of one unit flow. The flows delivered in a bundle must be separated or ‘unbundled’ at a sink node, and bundling and unbundling also incur costs. Therefore, unlike an ordinary network flow problem, our problem involves an additional decision for bundling. Although the FTNPB is a single commodity problem, it also has characteristics of a two commodity problem because two different transportation costs are associated with a unit flow depending on whether it is bundled or not. Note that finding an integral optimal solution in a two commodity network flow problem is NP-hard (Fortune et al., 1980), while a single commodity problem can be solved using an efficient network flow algorithm in polynomial time (Ahuja et al., 1993). Shintani et al. (2010) considered the FTNPB as an NP-hard integer programming problem and solved it using a general-purpose integer programming solver. However, Myung (2017) showed that when the underlying network has a star structure, the FTNPB can be transformed to a single commodity problem. Therefore, it is an interesting open question whether the FTNPB is NP-hard or not on an arbitrary network. Similarly, Myung and Moon (2014) showed that the seemingly two commodity model of relocating both standard and foldable containers by Moon et al. (2013) can be presented as a single commodity model. Both transformations by Myung (2017) and Myung and Moon (2014) commonly used an observation that a constraint for describing the numbers of flows that are bundled and unbundled could be represented as a flow conservation constraint. However, in addition to that observation, the following additional features of each model enable the two models to be transformed to corresponding single commodity network models. Due to the simple star structure considered, Myung (2017) could predetermine the optimal flows of one commodity, and Myung and Moon (2014) simplified the bundling decision via the assumption that all foldable containers should be folded (bundled) when transporting. Therefore, the techniques used Corresponding author. E-mail address: [email protected] (Y.-M. Yu). 1 The present research was conducted by the research fund of Dankook University in 2018. ⁎
https://doi.org/10.1016/j.tre.2019.101827 Received 2 April 2019; Received in revised form 28 November 2019; Accepted 11 December 2019 1366-5545/ © 2019 Elsevier Ltd. All rights reserved.
Transportation Research Part E 133 (2020) 101827
Y.-S. Myung and Y.-M. Yu
by Myung (2017) and Myung and Moon (2014) are not directly applicable to a single commodity formulation of the FTNPB on an arbitrary network. The contributions of this paper can be summarized as follows. First, we analyze the computational complexity of the FTNPB on an arbitrary network and prove that the problem is NP-hard. Secondly, because solving large-scale NP-hard problems is very time consuming, we develop an efficient heuristic to obtain near-optimal solutions within reasonable time. The key feature of our heuristic is that it uses a network flow algorithm. We also conduct intensive computing experiments to test our algorithm. Thirdly, the standard FTNPB assumes that a fixed number of flows are allowed to be bundled together, e.g., Shintani et al. (2010) used exactly four. We consider the variable bundling case where arbitrary numbers of flows can be bundled up to a given limit. We also show that the variable bundling version is still NP-hard and that a slight modification of our heuristic is directly applicable to this case. The remainder of the paper is organized as follows. In Section 2, we discuss some known and possible applications of the FTNPB and related studies. We formally describe the FTNPB and present an integer programming formulation of the problem in Section 3. Section 4 investigates the complexity of the problem, and in Section 5, we present a network flow algorithm-based heuristic to solve the proposed integer programming model. In Section 6, we consider the variable bundling version. We report the computational results in Section 7, and concluding remarks are presented in Section 8. 2. Applications and related studies In this section, we explain the application of the FTNPB in the empty foldable container relocation planning introduced by Shintani et al. (2010) and consider some possible applications in multimodal freight transportation. We also consider related studies. 2.1. Empty foldable container relocation Container transportation is a key component in current logistics, and allocating containers to meet demand is a challenging problem for a shipping company. As the demand and supply of empty containers are usually imbalanced at customer sites, a shipping company must reposition empty containers and consider the use of foldable containers to reduce transportation cost for repositioning. Recently, foldable containers have been produced by Holland Container Innovation, Korea Railway Research Institute, and various other companies. Usually, either 4 or 6 foldable containers can be folded and bundled and occupy the space of a single regular container. Although foldable containers reduce transportation cost, their usage also involves folding and unfolding costs. Therefore, the company must carefully consider whether using foldable containers to reduce the total cost of empty container relocation. After Konings and Thijs (2001) and Konings (2005) analyzed the potential cost savings of foldable containers, a variety of transportation models using foldable containers have been studied. Shintani et al. (2012), Moon et al. (2013), Myung and Moon (2014), and Moon and Hong (2016) considered multi-period container planning models for ocean transportation. These models include more complex decisions in addition to foldable container repositioning, such as purchasing and stocking containers. Zazgornik et al. (2012) proposed a combined model of vehicle routing and foldable container scheduling. Shintani et al. (2010) and Myung (2017) dealt with the problem of relocating empty containers in hinterland transportation. Foldable container reposition was combined with ship type decision-making by Wang et al. (2017), incorporated into drayage services by Zhang et al. (2018a), and into the intermodal transportation network of the Belt and Road Initiative by Zhang et al. (2018b). Other empty container allocation problems without considering foldable containers have been studied by Crainic et al. (1993), Shen and Khoong (1995), Cheang and Lim (2005), Li et al. (2007), Shintani et al. (2007), Dong and Song (2009), Song and Carter (2009), Moon et al. (2010), Song and Dong (2010), Song and Zhang (2010), Meng and Wang (2011), and Song and Dong (2011). As folding and unfolding can be represented as bundling and unbundling, the FTNPB is applicable to foldable container relocation. In fact, the models for foldable containers introduced by both Shintani et al. (2010) and Myung (2017) are equivalent to the FTNPB. Shintani et al. (2010) developed an integer programming formulation, and Myung (2017) showed that when the underlying network has star structure, the model can be solved using a network flow algorithm, i.e., solved in polynomial time. The FTNPB does not seem to be directly applicable to the other models with complex decisions combined with foldable container repositioning. However, the FTNPB is imbedded in those models and its properties can be used when analyzing the substructures of the models. 2.2. Multimodal freight transportation In this subsection, we consider possible applications of the FTNPB in multimodal freight transportation. We consider these applications in multimodal transportation because bundling/unbundling can represent the operations required for switching transportation modes. The scope of multimodal freight transportation is very broad, and a myriad of research has dealt with various multimodal freight transportation problems. Good surveys of multimodal freight transportation were conducted by Crainic and Kim (2007) and SteadieSeifi et al. (2014). The FTNPB can be applied to tactical and/or operational planning. The FTNPB is a single commodity problem and has a particular assumption that the transportation cost of flow depends not only on its amount but also whether it is bundled or not. However, most multimodal problems are modelled on multicommodity networks, and no models equivalent or similar to the FTNPB have been found in the literature. Here, we introduce a possible application of the FTNPB. In Korea, most land transportation companies provide shipment services using two types of trucks: one-ton trucks and four-ton trucks. Companies use one-ton trucks for the pickup process and four-ton trucks for long-distance transportation. To save transportation cost, companies can use both trucks in a transport chain by moving the cargos loaded in four one-ton trucks into one four-ton truck. This freight consolidation is a good option, especially in Korea, because four-ton 2
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trucks are restricted from passing through most city areas. The operation costs for the two types of trucks are similar, but loading and unloading incur costs. Therefore, Korean land transportation companies can obtain an optimal decision for their multimodal freight transportation by solving the FTNPB. Besides the above special instance, we can consider a more general situation where the FTNPB can be applied. Suppose that a company has multiple plants and multiple warehouses and a railway station exists near each plant and warehouse. The company can ship goods from a plant to a warehouse not only by truck but also by a truck-rail-truck multimodal chain. When the latter multimodal chain is chosen, goods are shipped from a plant to a nearby station by truck, moved by train between the two stations, and finally moved from the arrival station to a warehouse by truck. If we assume that the transshipment between truck and train incurs cost and that the transportation cost of the train is charged as a flat fee per cargo bay, which carries 3 or 4 times the amount of a unit truckload. Then, the FTNPB models the problem of determining minimum cost shipment of goods that satisfies the demands at warehouses from available supplies at plants. 3. Problem description and mathematical formulation In this section, we formally state the FTNPB and develop an integer programming formulation of the problem. Consider a directed network G = (N , A) that represents a transportation network. Each node in N is either a supply node or a demand node. Each arc corresponds to a transportation service and has an associated transportation cost. Let ND be the set of demand nodes and NS be the set of supply nodes, i.e., N = NS ND . Transshipment nodes, if they exist, are classified as supply nodes with zero supplies. Let Di be the demand of demand node i and Si be the supply of supply node i . The FTNPB aims to determine how to send flows from supply nodes to satisfy the demands of demand nodes at minimum cost. Unlike an ordinary single commodity network flow problem, we can send multiple flows, after grouping into a bundle, at the cost of one unit flow. We assume that no flows are given in a bundle at all supply nodes, and if the flows arrive in a bundle at demand nodes, they must be separated, i.e., unbundled. The bundling and unbundling also incur costs. Our objective is to minimize the total cost, which consists of transporting, bundling, and unbundling costs. We use the following notations to describe the cost coefficients:
Cij : unit transportation cost from node i to node j CiB : unit bundling cost at node i CiU : unit unbundling cost at node i The distinct characteristics of the FTNPB is that a certain number of flows can be grouped in a bundle and the transportation cost of one bundle on each arc is the same as that of a unit flow. We assume that b flows are grouped in a single bundle and thus the 1 transportation cost of a unit flow in a bundle from node i to node j is b Cij . We assume that the transportation cost of one bundle equal to that of a unit flow as in Shintani et al. (2010) and Myung (2017) but even in case of arbitrary transportation cost per bundle, the whole results of this study would not make any difference. We also assume that exactly b flows are grouped in a bundle as in Shintani et al. (2010) and Myung (2017). This restriction can be relaxed and we will consider the case where arbitrary number of flows are allowed to be bundled in Section 6. We assume that all the parameters are given as integers and that i NS Si = i ND Di . Now we describe the problem in mathematical form and use the following notations as decision variables:
x ij : amount of flows transported not in a bundle from node i to node j yij : number of bundles transported from node i to node j z iB : number of flows bundled at node i z iU : number of flows unbundled at node i Then, the total cost can be expressed as (i, j) A Cij (x ij + yij ) + i N (CiB z iB + CiU z iU ) , where the flows transported from node i to node j are represented by x ij + byij . Notice that at each node, if there are more outgoing bundles than incoming bundles, the bundling operation would be done, and if the opposite case occurs, the unbundling operation would be done. Therefore, z iB and z iU can be described as follows:
z iB = max b
yij {j (i, j) A}
z iU = max b
yji , 0 , { j (j , i ) A }
yji {j (j , i) A}
yij , 0 {j (i, j ) A}
Now, the integer programming formulation for the FTNPB is as follows:
(P)minimize
(CiB z iB + CiU z iU )
Cij (x ij + yij ) + (i, j ) A
i N
3
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subject to
(x ij + byij )
(xji + byji ) =
{j (i, j) A}
b
{ j (j , i ) A }
yji = z iB
yij {j (i, j ) A}
z iU
i
Si, if i NS Di , if i ND
(1)
N
(2)
{j (j, i) A}
x ij , yij , z iB , z iU
0 and integer
(i , j )
A, i
N
(3)
The constraints in (1) ensure flow conservation at each node. The constraints in (2) describe the relations among the variables. yij , z iB , and z iU have already been described. 4. Problem complexity In this section, we study the computational complexity of (P). As we already mentioned, Myung (2017) proved that if a given network is a star network, (P) can be solved in polynomial time. Here, we show that (P) is NP-hard on an arbitrary network. For this purpose, we introduce the following NP-complete decision version of 3-EXACT COVER (X3C): X3C
INSTANCE: A family F = {S1, , Sn} of three element subsets of S = {a1, QUESTION: Is there a subfamily of m subsets that covers S ?
, a3m} .
Theorem 1. (P) is NP-hard. Proof. We show that X3C is reducible to the decision version of (P). Given an instance of X3C, an instance of (P) is associated with it by the following procedure. We construct a transportation network that has 3m + n supply nodes and one demand node such that NS = NS1 NS2 , where NS1 = {1, , 3m} , NS2 = {3m + 1, , 3m + n} , and ND = {3m + n + 1} . Set Si = 1 for all i NS1, Si = 0 for all i NS2 , and D3m + n + 1 = 3m . We also establish the set of arcs in the network such that A = A1 A2 , where A1 = {(i, 3m + j) i NS1, 3m + j NS2, and ai Sj} and A2 = {(3m + i, 3m + n + 1) 3m + i NS2}. We assume that three flows are grouped in a bundle, i.e., b = 3. Set Cij = 0 for all (i, j) A1 and Cij = 1 for all (i, j) A2 . We also set CiB = CiU = 0 for all i N . Then, the answer to the X3C instance is yes if and only if the associated (P) has a feasible solution with cost m. □ 5. Solution method In this section, we develop a solution for the integer programming problem (P) presented in Section 2. As (P) is NP-hard, it would take huge CPU time to solve large-sized problems exactly. This was already observed by Shintani et al. (2010) and was proven to be true through our computing experiments that we present later. For this reason, we develop a heuristic to obtain near-optimal solutions within reasonable time. Our heuristic is based on the linear programming relaxation of (P), denoted by (LP), where the integrality condition in (3) is replaced with the non-negativity condition. Our algorithm first solves (LP) and fixes the values of the variables yij using the optimal solution of (LP). Then, we determine variables z iB and z iU through the constraints in (2) and the remaining variables x ij by solving the following subproblem (SP) of (P):
(SP) minimize
Cij xij (i, j) A
subject to
x ij {j (i, j ) A}
x ij
0 and integer
Si, if i
xji =
D i , if i
{j (j, i) A}
(i , j )
NS (4)
ND
(5)
A
{
}
{
}
byij byji and Di = Di + byij byji . where Si = Si {j (i, j) A} { j (j , i ) A } {j (i, j ) A} {j (j , i) A} The main concept behind our heuristic is based on the observation that both (LP) and (SP) can be viewed as network flow problems. It is straightforward that (SP) is a network flow problem because the constraints in (4) are the flow conservation constraints. However, it is non-trivial to show that we can solve (LP) using a network flow algorithm. We formally verify our findings as follows: Lemma 1. (LP) can be transformed to a network flow problem. Proof. The constraints in (1) can be modified to the following constraint using (2):
x ji + z iB
x ij {j (i, j) A}
{j (j, i) A}
ziU =
Si, if i NS Di , if i ND
(1′)
Replacing the term byij with fij , (LP) can be transformed into the following equivalent problem: 4
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Y.-S. Myung and Y.-M. Yu
Fig. 1. Decision variables corresponding to arcs.
(LP') minimize
Cij x ij + (i, j ) A
subject to
x ij , fij
z iU =
{j (j, i) A}
f ji = z iB
fij
(CiB z iB + CiU ziU ) i N
xji + z iB
x ij {j (i, j) A}
{j (i, j) A}
1 f + b ij
z iU
i
Si, if i NS Di , if i ND
N
(2′)
{ j (j , i ) A }
0
(i , j )
(1′)
A
Now we show that (LP') is a network flow problem. The constraints in (1') and (2') can be seen as the flow conservation constraints in a network. The corresponding network is shown in Fig. 1, where node i and node i' correspond to the constraints in (1') and (2'), respectively, and each arc corresponds to the variable labeled on the arc. □ As (LP) and (SP) are network flow problems, both problems can be solved using an efficient network flow algorithm instead of a general-purpose linear and/or integer programming method. More details on network flow problems and algorithms are given by Ahuja et al. (1993). As we mentioned, our algorithm first solves (LP') and determines the values of yij using the optimal solution of (LP'). An optimal solution of (LP') satisfies integrality and is also optimal to (P) if the optimal values of variables fij are multiples of b. In that case, yij = 1 bfij for each (i, j) A, and the optimal values of the remaining variables x ij , z iB , and z iU from (LP') are also optimal for (P). On the other hand, if any fij is not a multiple of b, we set yij = 1 bfij , determine the variables z iB and z iU through the constraints in (2), and solve (SP) to obtain the values of the variables x ij . 6. Variable bundling Up to now, we have assumed that exactly b flows are grouped in a bundle. However, if we allow arbitrary numbers of flows to be bundled, the resulting FTNPB would be more versatile and could be used in more applications. In this section, we consider the extended model, which we call the FTNPB with variable bundling, where up to b flows can be bundled. We analyzed the complexity of the extended model and show that our heuristic is directly applicable to the problem. 6.1. Problem complexity The FTNPB with variable bundling is still NP-hard. To prove this, we can use the proof of Theorem 1. Consider the same transportation network in the proof; the X3C instance is yes if and only if the FTNPB with variable bundling has a feasible solution with cost m. 6.2. Solution method First, we develop the mathematical programming formulation of the variable bundling version. As we can bundle 2, 3, , or b flows, we require the variables yijk for 2 k b instead of yij to represent the number of bundles grouping k flows. Then, the new problem can be formulated as follows:
Cij (xij + yij2 +
(PV) minimize (i, j) A
+ yijb ) +
(CiB z iB + CiU ziU ) i N
5
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Y.-S. Myung and Y.-M. Yu
subject to
{j (i, j ) A}
(x ij + 2yij2 +
+ byijb )
(2yji2 +
{j (i, j) A}
{j (j, i) A}
(xji + 2yji2 +
+byjib )
Si, if i NS Di , if i ND
=
(2yij2 +
+ byijb )
(6)
+ byjib ) = z iB
ziU
i
N
(7)
{ j (j , i ) A }
x ij , yijk , ziB , ziU
0 and integer
(i , j )
A, i
N, 2
k
b
(8)
We show that our heuristic developed in Section 5 can solve (PV) by a slight modification. Like the heuristic for (P), our new heuristic solves the linear programming relaxation of (PV), denoted by (LPV), determines yijk with 2 k b by truncating the
fractional solution of (LPV), sets z iB and z iU through the constraints in (7), and solves the following subproblem (SPV) to find x ij :
(SPV) minimize
Cij xij (i, j) A
subject to
x ij {j (i, j ) A}
x ij
D i , if i
{j (j, i) A}
0 and integer
{
Si, if i
xji = (i , j )
NS (9)
ND
(10)
A
}
{
}
+byjib ) . where Si = Si and Di = Di + (SPV) is exactly the same as (SP) and is a network flow problem. All that remains is to show that (LPV) can be solved using a network flow algorithm as for (LP). This can be explained in the same way as the proof of Lemma 1. We restate it as follows:
(2yij2 + {j (i, j) A}
+ byijb )
(2yji2 + {j (j, i) A}
+ byjib )
(2yij2 + {j (i, j) A}
+byijb )
(2yji2 + {j (j, i ) A}
Lemma 2. (LPV) can be transformed to a network flow problem. Proof. The constraints in (6) can be modified to the following constraint using (7):
x ji + z iB
x ij {j (i, j) A}
Replacing
{j (j, i) A}
kyijk
with
fijk
for 2
(LPV') minimize
Cij x ij +
subject to
x ij , fijk
(fij2 +
0
+ fijb )
(i , j )
1 2 f + 2 ij
1 + fijb + b
xji + z iB
x ij {j (i, j) A}
z iU =
{j (j, i) A}
{j (j, i) A}
A, 2
(6′)
b , (LVP) can be transformed into the following equivalent problem:
k
(i, j ) A
{j (i, j) A}
Si, if i NS Di , if i ND
ziU =
(f ji2 +
k
+f jib ) = z iB
(CiB z iB + CiU z iU ) i N
Si, if i NS Di , if i ND z iU
i
N
(6′) (7′)
b
As in (LP'), the constraints in (6' ) and (7') can be seen as the flow conservation constraints in a network. The corresponding network is the same as the network shown in Fig. 1 except that there are multiple edges from node i to node j , each of which corresponds to fijk . □ 7. Computational results In this section, we describe and report the results of computing experiments to evaluate the performance of our heuristic and to analyse the effects of bundling. We also conduct the sensitivity analysis to find how the changes of cost parameters affect the total cost. 7.1. Performance of our heuristic For intensive tests, we randomly generated data instances with various sizes. We first drew a 100 × 100 rectangle on which node sites were randomly located, and networks were generated with various numbers of nodes. Arcs were also randomly selected until the resulting network density (ratio of generated arcs to total number of possible arcs) reached the predetermined level (20, 35, or 50%). We generated 5 instances for each combination of a number of nodes and network density. The cost parameters for transportation, bundling, and unbundling were set as in Shintani et al. (2010). We assumed that a fixed number of flows was allowed to be bundled and set b = 4 . Test runs were performed on a PC (2.4 GHz and 8G RAM). 6
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Table 1 Computational results for small-sized problems. No. of nodes
10
Network density
0.2
0.35
0.5
15
0.2
0.35
0.5
20
0.2
0.35
0.5
z(H)
z(P)
z(LP)
z (H ) z (P )
z (H ) z (LP )
z (P ) z (LP )
Computation time (seconds) Heuristic
Optimal
23693.0 27954.0 21511.0 25395.0 22483.0 20452.0 29653.0 20150.0 14941.0 23786.0 17401.0 23034.0 14480.0 23093.0 21069.0
23398.0 27615.0 21434.0 25252.0 22483.0 20452.0 29197.0 18812.0 13807.0 22898.0 15784.0 21673.0 13543.0 22244.0 20283.0
21790.8 26082.0 20552.8 23162.5 21183.2 19540.2 28106.5 17758.8 12683.8 21584.0 14868.0 20900.5 12403.5 21400.8 19525.3
1.01 1.01 1.00 1.01 1.00 1.00 1.02 1.07 1.08 1.04 1.10 1.06 1.07 1.04 1.04
1.09 1.07 1.05 1.10 1.06 1.05 1.06 1.13 1.18 1.10 1.17 1.10 1.17 1.08 1.08
1.07 1.06 1.04 1.09 1.06 1.05 1.04 1.06 1.09 1.06 1.06 1.04 1.09 1.04 1.04
0.031 0.016 0.015 0.016 0.031 0.016 0.016 0.015 0.016 0.015 0.016 0.015 0.031 0.016 0.031
0.047 0.047 0.016 0.063 0.203 0.094 0.110 0.016 0.062 0.062 0.047 0.047 0.156 0.047 0.047
29278.0 32269.0 29806.0 30586.0 25319.0 34158.0 29161.0 19041.0 25041.0 21648.0 25182.0 21649.0 30797.0 22810.0 28455.0
28297.0 31582.0 28018.0 30513.0 24886.0 33163.0 27777.0 17447.0 24624.0 20944.0 22937.0 20585.0 29336.0 21582.0 27076.0
26422.0 30405.8 26207.5 28675.5 23026.5 31568.8 26724.0 16315.0 23246.8 19935.8 21737.3 19460.3 28192.8 20363.8 25745.8
1.03 1.02 1.06 1.00 1.02 1.03 1.05 1.09 1.02 1.03 1.10 1.05 1.05 1.06 1.05
1.11 1.06 1.14 1.07 1.10 1.08 1.09 1.17 1.08 1.09 1.16 1.11 1.09 1.12 1.11
1.07 1.04 1.07 1.06 1.08 1.05 1.04 1.07 1.06 1.05 1.06 1.06 1.04 1.06 1.05
0.031 0.031 0.015 0.031 0.016 0.016 0.016 0.015 0.015 0.031 0.031 0.015 0.016 0.031 0.031
0.063 0.031 0.063 0.063 0.047 0.141 0.062 0.047 0.093 0.047 0.047 0.078 0.219 0.031 0.031
31500.0 33834.0 30400.0 33577.0 37158.0 29720.0 26371.0 32871.0 31585.0 29310.0 32175.0 38664.0 27187.0 38727.0 33645.0
29529.0 31783.0 30013.0 31426.0 35072.0 28106.0 25331.0 31547.0 30163.0 26371.0 31506.0 37001.0 26407.0 37537.0 32049.0
27712.5 29559.0 28324.8 29644.8 32597.5 26481.3 23676.3 30068.5 28623.3 25052.0 29975.0 35383.3 25190.3 36278.8 30583.5
1.07 1.06 1.01 1.07 1.06 1.06 1.04 1.04 1.05 1.11 1.02 1.04 1.03 1.03 1.05
1.14 1.14 1.07 1.13 1.14 1.12 1.11 1.09 1.10 1.17 1.07 1.09 1.08 1.07 1.10
1.07 1.08 1.06 1.06 1.08 1.06 1.07 1.05 1.05 1.05 1.05 1.05 1.05 1.03 1.05
0.031 0.015 0.015 0.015 0.031 0.031 0.015 0.031 0.031 0.015 0.015 0.016 0.031 0.016 0.063
0.187 0.078 0.218 0.031 0.047 0.156 0.031 0.062 0.140 0.093 0.062 0.063 0.235 0.109 0.047
We classified the data instances into three groups. The first group included relatively small-sized data instances with 10, 15, and 20 nodes; the computational results for this group are summarized in Table 1. The second group consisted of medium-sized problems with 30, 50, and 100 nodes (Table 2) and the last group was large-sized problems with 150, 200, and 300 nodes (Table 3). For each group of problems, we evaluated the performance of our algorithm by comparing the objective value of our heuristic (z(H)) with the objective value obtained from (P) (z(P)). We set a time limit of 100 s, and we could not exactly solve (P) for several large-sized problems within the time limit. Therefore, we also provide the objective value obtained from (LP) (z(LP)), which is a lower bound of z (P). In each table, we report the computation times for our heuristic and an integer programming algorithm. When solving (LP), we used a general-purpose linear programming solver instead of a network flow algorithm for computational convenience. Nevertheless, the computational advantages of our heuristic over an integer programming algorithm are clearly reflected in the results. We used optimization software CPLEX 12.9 to solve both an integer and linear programming problem. The results revealed that our heuristic finds near-optimal solutions for fairly large problems within reasonable time. Note that in all instances, the linear programming relaxation does not provide an integer solution but provides very good ratios of z(LP) to z(P). This good integrality gap seems to be due to the network structure of the FTNPB. For each combination of number of nodes and network density, most of the average ratios of z(H) to z(P) are within 5%, and the ratios of z(H) to z(LP) are within 10%. Also, note that the computation times of our heuristic to solve the large-sized problems in Table 3 are within a couple of seconds, while the 7
Transportation Research Part E 133 (2020) 101827
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Table 2 Computational results for medium-sized problems. No. of nodes
30
Network density
z(H)
0.2
42465.0 53426.0 48169.0 47581.0 55823.0 43072.0 44468.0 39998.0 53901.0 60008.0 47188.0 53264.0 42098.0 39177.0 44825.0
40673.0 49817.0 46092.0 46328.0 55287.0 41166.0 41095.0 37540.0 52023.0 57460.0 44698.0 50945.0 40302.0 36906.0 42368.0
38488.3 47752.5 43460.5 43521.3 52766.0 38639.0 38761.3 35816.3 49998.5 55548.3 42963.5 48984.3 38391.8 35008.3 40514.0
1.04 1.07 1.05 1.03 1.01 1.05 1.08 1.07 1.04 1.04 1.06 1.05 1.04 1.06 1.06
1.10 1.12 1.11 1.09 1.06 1.11 1.15 1.12 1.08 1.08 1.10 1.09 1.10 1.12 1.11
72063.0 90595.0 78435.0 81737.0 69791.0 78205.0 81875.0 64877.0 71617.0 82246.0 73350.0 68071.0 71409.0 64988.0 68726.0
69412.0 86847.0 74728.0 77399.0 65696.0 75929.0 79231.0 61896.0 68892.0 79842.0 69494.0 64465.0 67909.0 63140.0 64445.0
65570.5 83290.3 70823.5 73287.8 62102.0 72372.3 76404.0 58964.0 65950.8 76546.8 66641.8 61683.8 64940.0 60386.8 61348.0
1.04 1.04 1.05 1.06 1.06 1.03 1.03 1.05 1.04 1.03 1.06 1.06 1.05 1.03 1.07
163470.0 152857.0 159611.0 135936.0 133749.0 147771.0 133777.0 131094.0 142768.0 137010.0 157365.0 136342.0 147947.0 130119.0 155289.0
154228.0 145578.0 151723.0 129649.0 126214.0 140409.0 128661.0 123937.0 136850.0 130688.0 151256.0* 128713.0 141775.0 123922.0 145895.0
146766.0 139241.3 145262.0 123445.3 119112.5 134255.0 122353.3 118400.0 130717.8 124864.8 146525.3 122727.0 136273.3 117847.3 140249.3
1.06 1.05 1.05 1.05 1.06 1.05 1.04 1.06 1.04 1.05 1.04 1.06 1.04 1.05 1.06
0.35
0.5
50
0.2
0.35
0.5
100
0.2
0.35
0.5
z(P)
z(LP)
z (H ) z (P )
z (H ) z (LP )
z (P ) z (LP )
Computation time (seconds) Heuristic
Optimal
1.06 1.04 1.06 1.06 1.05 1.07 1.06 1.05 1.04 1.03 1.04 1.04 1.05 1.05 1.05
0.047 0.015 0.016 0.032 0.031 0.031 0.031 0.047 0.031 0.047 0.047 0.047 0.047 0.047 0.047
0.328 0.109 0.063 1.125 0.156 0.078 0.078 0.063 0.047 0.109 0.297 0.094 0.125 0.079 0.062
1.10 1.09 1.11 1.12 1.12 1.08 1.07 1.10 1.09 1.07 1.10 1.10 1.10 1.08 1.12
1.06 1.04 1.06 1.06 1.06 1.05 1.04 1.05 1.04 1.04 1.04 1.05 1.05 1.05 1.05
0.063 0.047 0.047 0.047 0.047 0.062 0.078 0.078 0.078 0.094 0.093 0.094 0.094 0.110 0.093
0.422 0.970 0.328 0.359 0.109 1.343 0.375 0.344 0.156 0.406 0.484 1.547 1.594 0.141 0.219
1.11 1.10 1.10 1.10 1.12 1.10 1.09 1.11 1.09 1.10 1.07 1.11 1.09 1.10 1.11
1.05 1.05 1.04 1.05 1.06 1.05 1.05 1.05 1.05 1.05 1.03 1.05 1.04 1.05 1.04
0.171 0.156 0.172 0.187 0.172 0.250 0.250 0.265 0.266 0.250 0.360 0.359 0.375 0.344 0.359
1.063 1.203 1.172 0.750 0.891 1.110 3.125 1.329 1.281 4.532 > 100 0.828 7.937 0.781 2.828
* Best solution obtained within the time limit.
integer programming solver exceeded the time limit of 100 s for half of the problems. 7.2. Effects of bundling In order to investigate the effects of bundling, we conducted the experiments of comparing the total costs of the three FTNP models: a model with fixed bundling; a model with variable bundling; and a model without bundling. The results given in Table 4 imply that bundling saves the total cost much and that additional cost saving by variable bundling is not that much. 7.3. Sensitivity analysis on cost parameters We also conducted the sensitivity analysis to show how the changes of transportation cost and folding/unfolding cost affect the total cost. The results given in Table 5 revealed that the total cost increases almost at the same rate as transportation cost when bundling is not allowed but increases at lower late when bundling is allowed. In Table 6, the results of the sensitivity analysis on folding/unfolding cost are also provided. 8
Transportation Research Part E 133 (2020) 101827
Y.-S. Myung and Y.-M. Yu
Table 3 Computational results for large-sized problems. No. of nodes
150
Network density
z(H)
0.2
204688.0 207595.0 205149.0 222667.0 230598.0 208490.0 194802.0 200115.0 226844.0 221712.0 215497.0 230988.0 211669.0 203931.0 205639.0
194657.0 197335.0 194895.0 214183.0 221541.0 198671.0 186658.0 192276.0 214968.0 212666.0 205513.0 218942.0 202570.0 193738.0 195133.0
185738.8 187784.5 186010.3 206222.3 212553.3 188931.8 178229.8 184203.8 207092.3 204433.5 196842.3 210081.5 194616.8 185799.8 187501.0
1.05 1.05 1.05 1.04 1.04 1.05 1.04 1.04 1.06 1.04 1.05 1.06 1.04 1.05 1.05
1.10 1.11 1.10 1.08 1.08 1.10 1.09 1.09 1.10 1.08 1.09 1.10 1.09 1.10 1.10
292479.0 291000.0 273744.0 293856.0 280506.0 266088.0 301675.0 250734.0 259594.0 264132.0 265419.0 272746.0 303751.0 247738.0 279311.0
276275.0 277528.0 260047.0 279763.0 267197.0 253199.0 285727.0* 238093.0 245313.0 252248.0 251305.0 260688.0 289935.0 237919.0 265892.0
265297.0 265987.5 248241.0 268203.3 255556.0 241259.0 274043.8 227006.5 233126.3 240845.8 239619.3 249617.5 278264.5 226886.0 254040.0
1.06 1.05 1.05 1.05 1.05 1.05 1.06 1.05 1.06 1.05 1.06 1.05 1.05 1.04 1.05
406073.0 375927.0 381221.0 427440.0 414184.0 376241.0 380873.0 385992.0 414638.0 387404.0 425470.0 417038.0 423032.0 386860.0 393710.0
387399.0 358847.0 361197.0 407240.0 393919.0 359180.0* 365228.0 368624.0* 399088.0 370808.0* 404653.0* 395690.0* 406578.0* 369521.0* 375706.0*
370554.5 341852.5 344188.5 389028.0 376806.5 343185.8 349269.8 350809.0 382294.8 354757.5 388347.8 378426.8 390596.3 353311.0 360564.8
1.05 1.05 1.06 1.05 1.05 1.05 1.04 1.05 1.04 1.04 1.05 1.05 1.04 1.05 1.05
0.35
0.5
200
0.2
0.35
0.5
300
0.2
0.35
0.5
z(P)
z(LP)
z (H ) z (P )
z (H ) z (LP )
z (P ) z (LP )
Computation time (seconds) Heuristic
Optimal
1.05 1.05 1.05 1.04 1.04 1.05 1.05 1.04 1.04 1.04 1.04 1.04 1.04 1.04 1.04
0.391 0.360 0.391 0.359 0.375 0.562 0.563 0.562 0.578 0.578 0.781 0.797 0.797 0.812 0.781
14.016 1.515 1.672 6.750 2.110 10.281 15.453 20.157 14.422 20.281 23.125 16.969 26.890 1.891 23.922
1.10 1.09 1.10 1.10 1.10 1.10 1.10 1.10 1.11 1.10 1.11 1.09 1.09 1.09 1.10
1.04 1.04 1.05 1.04 1.05 1.05 1.04 1.05 1.05 1.05 1.05 1.04 1.04 1.05 1.05
0.594 0.719 0.594 0.594 0.594 0.969 1.047 1.000 1.015 0.984 1.422 1.454 1.484 1.438 1.469
2.938 3.016 17.375 3.282 11.766 28.469 > 100 29.969 32.313 9.781 44.375 61.375 51.484 34.078 52.438
1.10 1.10 1.11 1.10 1.10 1.10 1.09 1.10 1.08 1.09 1.10 1.10 1.08 1.09 1.09
1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.04 1.05 1.04 1.05 1.04 1.05 1.04
1.375 1.343 1.344 1.406 1.375 2.375 2.469 2.500 2.468 2.484 3.781 3.532 3.735 3.641 3.500
62.671 48.141 11.859 19.484 37.046 > 100 58.907 > 100 91.094 > 100 > 100 > 100 > 100 > 100 > 100
* Best solution obtained within the time limit.
8. Conclusions In this paper, we considered the freight transportation network problem with the bundling option. In this model, when we choose the bundling option, we can send multiple goods, after grouping into a bundle, at the cost of one unit good, but there are costs for bundling and unbundling. Our problem is to decide how to send goods with an additional decision for bundling from suppliers to customers at minimum cost. The problem originally arose from relocating empty containers in hinterland transportation, but it can also be applied to multimodal freight transportation. We have shown that the problem is NP-hard and that it is time consuming to solve a large-sized problem exactly using an integer programming algorithm. We found that the problem has a network flow problem structure, and we developed an efficient heuristic that uses a network flow algorithm. We also presented the results of computational experiments and showed that our heuristic finds near-optimal solutions even for fairly large problems within a reasonable time.
9
Transportation Research Part E 133 (2020) 101827
Y.-S. Myung and Y.-M. Yu
Table 4 Cost comparison among different bundling strategies. No. of nodes
Network density
50
No bundling(a)
0.2
150298.0 183545.0 163874.0 187395.0 130080.0 135717.0 147420.0 128992.0 138591.0 151199.0 138827.0 128555.0 131116.0 120379.0 130460.0
0.35
0.5
Fixed bundling (4 flows)
Variable bundling (2 3 4 flows)
(b)
(b/a)
(c)
(c/a)
69412.0 86847.0 74728.0 77399.0 65696.0 75929.0 79231.0 61896.0 68892.0 79842.0 69494.0 64465.0 67909.0 63140.0 64445.0
0.46 0.47 0.46 0.41 0.51 0.56 0.54 0.48 0.50 0.53 0.50 0.50 0.52 0.52 0.49
67540.0 85413.0 73026.0 75214.0 64002.0 74264.0 78199.0 60446.0 67576.0 78273.0 68269.0 63133.0 66480.0 62076.0 62937.0
0.45 0.47 0.45 0.40 0.49 0.55 0.53 0.47 0.49 0.52 0.49 0.49 0.51 0.52 0.48
Table 5 Sensitivity analysis on the increases of transportation cost. (No bundling case) No. of nodes
50
Network density
0.2
0.35
0.5
Original data (a)
150298.0 183545.0 163874.0 187395.0 130080.0 135717.0 147420.0 128992.0 138591.0 151199.0 138827.0 128555.0 131116.0 120379.0 130460.0
10% increase
20% increase
30% increase
40% increase
(b)
(c)
(b)
(c)
(b)
(c)
(b)
(c)
165345.0 201904.0 180349.0 206206.0 143113.0 149388.0 162189.0 142108.0 152476.0 166486.0 152727.0 141499.0 144280.0 132495.0 143627.0
10.0% 10.0% 10.1% 10.0% 10.0% 10.1% 10.0% 10.2% 10.0% 10.1% 10.0% 10.1% 10.0% 10.1% 10.1%
180401.0 220285.0 196750.0 225136.0 156150.0 162969.0 176937.0 154990.0 166362.0 181616.0 166658.0 154364.0 157422.0 144490.0 156642.0
20.0% 20.0% 20.1% 20.1% 20.0% 20.1% 20.0% 20.2% 20.0% 20.1% 20.0% 20.1% 20.1% 20.0% 20.1%
195465.0 238759.0 213238.0 243839.0 169257.0 176622.0 191727.0 167915.0 180252.0 196699.0 180569.0 167274.0 170578.0 156570.0 169740.0
30.1% 30.1% 30.1% 30.1% 30.1% 30.1% 30.1% 30.2% 30.1% 30.1% 30.1% 30.1% 30.1% 30.1% 30.1%
210501.0 257147.0 229605.0 262666.0 182319.0 190180.0 206550.0 180919.0 194202.0 211893.0 194514.0 180248.0 183692.0 168706.0 182942.0
40.1% 40.1% 40.1% 40.2% 40.2% 40.1% 40.1% 40.3% 40.1% 40.1% 40.1% 40.2% 40.1% 40.1% 40.2%
(Fixed bundling case) No. of nodes
50
Network density
0.2
0.35
0.5
(c) =
(b ) (a) (a)
Original data (a)
69412.0 86847.0 74728.0 77399.0 65696.0 75929.0 79231.0 61896.0 68892.0 79842.0 69494.0 64465.0 67909.0 63140.0 64445.0
10% increase
20% increase
30% increase
40% increase
(b)
(c)
(b)
(c)
(b)
(c)
(b)
(c)
63260.0 83219.0 71820.0 74791.0 62808.0 72061.0 75299.0 59324.0 65768.0 76026.0 66382.0 61477.0 64610.0 60024.0 61633.0
6.1% 5.8% 6.2% 6.7% 5.7% 5.2% 5.2% 5.9% 5.6% 5.4% 5.6% 5.6% 5.4% 5.5% 5.7%
58903.0 79599.0 68746.0 71915.0 59780.0 68339.0 71547.0 56602.0 62792.0 72064.0 63186.0 58561.0 61391.0 57180.0 58777.0
12.2% 11.6% 12.3% 13.4% 11.3% 10.3% 10.3% 11.7% 11.2% 10.7% 11.2% 11.2% 10.8% 10.9% 11.4%
54823.0 75503.0 65690.0 69356.0 56848.0 64559.0 67515.0 53842.0 59524.0 68332.0 60040.0 55673.0 58237.0 54122.0 55892.0
18.3% 17.5% 18.4% 20.0% 17.0% 15.5% 15.5% 17.4% 16.9% 16.0% 16.9% 16.8% 16.2% 16.4% 17.2%
51087.0 71883.0 62970.0 66752.0 53912.0 60715.0 63423.0 51066.0 56492.0 64324.0 56800.0 52657.0 54957.0 51242.0 53036.0
24.4% 23.3% 24.6% 26.7% 22.6% 20.6% 20.7% 23.2% 22.5% 21.3% 22.5% 22.4% 21.5% 21.8% 22.9%
× 100(%) .
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Transportation Research Part E 133 (2020) 101827
Y.-S. Myung and Y.-M. Yu
Table 6 Sensitivity analysis on the decreases of folding and unfolding cost. No. of nodes
50
Network density
0.2
0.35
0.5
(c) =
(a) (b ) (a)
Original data (a)
69412.0 86847.0 74728.0 77399.0 65696.0 75929.0 79231.0 61896.0 68892.0 79842.0 69494.0 64465.0 67909.0 63140.0 64445.0
10% decrease
20% decrease
30% decrease
40% decrease
(b)
(c)
(b)
(c)
(b)
(c)
(b)
(c)
63260.0 83219.0 71820.0 74791.0 62808.0 72061.0 75299.0 59324.0 65768.0 76026.0 66382.0 61477.0 64610.0 60024.0 61633.0
8.9% 4.2% 3.9% 3.4% 4.4% 5.1% 5.0% 4.2% 4.5% 4.8% 4.5% 4.6% 4.9% 4.9% 4.4%
58903.0 79599.0 68746.0 71915.0 59780.0 68339.0 71547.0 56602.0 62792.0 72064.0 63186.0 58561.0 61391.0 57180.0 58777.0
15.1% 8.3% 8.0% 7.1% 9.0% 10.0% 9.7% 8.6% 8.9% 9.7% 9.1% 9.2% 9.6% 9.4% 8.8%
54823.0 75503.0 65690.0 69356.0 56848.0 64559.0 67515.0 53842.0 59524.0 68332.0 60040.0 55673.0 58237.0 54122.0 55892.0
21.0% 13.1% 12.1% 10.4% 13.5% 15.0% 14.8% 13.0% 13.6% 14.4% 13.6% 13.6% 14.2% 14.3% 13.3%
51087.0 71883.0 62970.0 66752.0 53912.0 60715.0 63423.0 51066.0 56492.0 64324.0 56800.0 52657.0 54957.0 51242.0 53036.0
26.4% 17.2% 15.7% 13.8% 17.9% 20.0% 20.0% 17.5% 18.0% 19.4% 18.3% 18.3% 19.1% 18.8% 17.7%
× 100(%) .
CRediT authorship contribution statement Young-Soo Myung: Conceptualization, Methodology, Writing - original draft. Yung-Mok Yu: Supervision, Data curation, Software, Writing - review & editing. References Ahuja, R.K., Magnanti, T.L., Orlin, J.B., 1993. Network Flows, Prentice-Hall, New Jersey, USA. Cheang, B., Lim, A., 2005. A network flow bases method for the distribution of empty containers. Int. J. Comput. Appl. Technol. 22, 198–204. Crainic, T.G., Gendreau, M., Dejax, P., 1993. Dynamic and stochastic models for the allocation of empty containers. Oper. Res. 41, 102–126. Crainic, T.G., Kim, K., 2007. Intermodal transportation. Handbooks Oper. Res. Manage. Sci. 14, 467–537. Dong, J.X., Song, D.P., 2009. Container fleet sizing and empty repositioning in liner shipping systems. Transp. Res. Part E 45, 860–877. Fortune, S., Hopcroft, J., Wyllie, J., 1980. The directed subgraph homeomorphism problem. Theoret. Comput. Sci. 10, 111–121. Konings, R., Thijs, R., 2001. Foldable containers: a new perspective on reducing container-repositioning costs. Eur. J. Oper. Res. 1 (4), 333–352. Konings, R., 2005. Foldable containers to reduce the costs of empty transport? A cost–benefit analysis from a chain and multi-actor perspective. Marit. Econ. Logist. 7 (3), 223–249. Li, J.A., Leung, S.C.H., Wu, Y., Liu, K., 2007. Allocation of empty containers between multi-ports. Eur. J. Oper. Res. 182, 400–412. Meng, Q., Wang, L., 2011. Liner shipping service network design with empty container repositioning. Transp. Res. Part E 47, 605–708. Moon, I.K., Do Ngoc, A.D., Hur, Y.S., 2010. Positioning empty containers among multiple ports with leasing and purchasing consideration. OR Spectrum 32, 765–786. Moon, I.K., Do Ngoc, A.D., Konings, R., 2013. Foldable and standard containers in empty container repositioning. Transp. Res. Part E 49, 107–124. Moon, I.K., Hong, H.J., 2016. Repositioning of empty containers using both standard and foldable containers. Marit. Econ. Logist. 18 (1), 61–77. Myung, Y.S., Moon, I.K., 2014. A network flow model for the optimal allocation of both foldable and standard containers. Oper. Res. Lett. 42, 484–488. Myung, Y.S., 2017. Efficient solution methods for the integer programming models of relocating empty containers in the hinterland transportation network. Transp. Res. Part E 108, 52–59. Shen, W.S., Khoong, C.M., 1995. A DSS for empty container distribution planning. Decis. Support Syst. 15, 75–82. Shintani, K., Konings, R., Imai, A., 2010. The impact of foldable containers on container fleet management costs in hinterland transport. Transp. Res. Part E 46, 750–763. Shintani, K., Konings, R., Imai, A., 2012. The impact of foldable containers on the costs of container fleet management in liner shipping networks. Marit. Econ. Logist. 14, 455–479. Shintani, K., Imai, A., Nishimura, E., 2007. The container shipping network design problem with empty container repositioning. Transp. Res. Part E 43, 39–59. Song, D.P., Carter, J., 2009. Empty container repositioning in liner shipping. Marit. Policy Manage. 36, 291–307. Song, D.P., Dong, J.X., 2010. Effectiveness of an empty container repositioning policy with flexible destination ports. Transp. Pol. 18, 92–101. Song, D.P., Dong, J.X., 2011. Flow balancing-based empty container repositioning in typical shipping service routes. Marit. Econ. Logist. 13, 61–77. Song, D.P., Zhang, Q., 2010. A fluid flow model for empty container repositioning policy with a single port and stochastic demand. Siam J. Control Optim. 48 (5), 3623–3642. SteadieSeifi, M., Dellaert, N.P., Nuijten, W., Woensel, T.V., Raoufi, R., 2014. Multimodal freight transportation planning: a literature review. Eur. J. Oper. Res. 233, 1–15. Wang, K., Wang, S., Zhen, L., Qu, X., 2017. Ship type decision considering empty container repositioning and foldable containers. Transp. Res. Part E 108, 97–121. Zazgornik, J., Gronalt, M., Hirsch, P., 2012. The combined vehicle routing and foldable container scheduling problem: a model formulation and tabu search bases solution approaches. INFOR 30, 147–162. Zhang, R., Zhao, H., Moon, I.K., 2018a. Range-based truck-state transition modeling method for foldable container drayage services. Transp. Res. Part E 118, 225–239. Zhang, S., Ruan, X., Xia, Y., Feng, X., 2018b. Foldable container in empty container repositioning in intermodal transportation network of Belt and Road Initiative: strengths and limitations. Maritime Poli. Manage. 45, 351–369.
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Transportation Research Part E journal homepage: www.elsevier.com/locate/tre
Freight transportation network model with bundling option Young-Soo Myung, Yung-Mok Yu
⁎,1
T
Department of Business Administration, Dankook University, Yongin-si, Gyeonggi-do 16890, Republic of Korea
ARTICLE INFO
ABSTRACT
Keywords: Network flow Bundling option Relocating foldable containers
We consider a freight transportation network model in which we can send goods with or without bundling. When we choose the bundling option, we can send multiple goods, after grouping into a bundle, at the cost of one unit good; however, bundling and unbundling also incur costs. The model was originally developed to solve the problem of relocating empty containers in hinterland transportation, but it also has an application in multimodal freight transportation. We investigate the complexity of the problem and develop an efficient heuristic that uses a network flow algorithm.
1. Introduction In this paper, we consider the Freight Transportation Network Problem with the Bundling option (FTNPB), which was proposed by Shintani et al. (2010) to address the problem of relocating empty containers in hinterland transportation. The FTNPB is a variant of the single commodity network flow problem with multiple sources and sinks. The problem involves sending flows from a set of source nodes to satisfy the demands at the sink nodes at minimum cost, but unlike in a standard problem, we can group a certain number of flows into a bundle and send each bundle at the cost of one unit flow. The flows delivered in a bundle must be separated or ‘unbundled’ at a sink node, and bundling and unbundling also incur costs. Therefore, unlike an ordinary network flow problem, our problem involves an additional decision for bundling. Although the FTNPB is a single commodity problem, it also has characteristics of a two commodity problem because two different transportation costs are associated with a unit flow depending on whether it is bundled or not. Note that finding an integral optimal solution in a two commodity network flow problem is NP-hard (Fortune et al., 1980), while a single commodity problem can be solved using an efficient network flow algorithm in polynomial time (Ahuja et al., 1993). Shintani et al. (2010) considered the FTNPB as an NP-hard integer programming problem and solved it using a general-purpose integer programming solver. However, Myung (2017) showed that when the underlying network has a star structure, the FTNPB can be transformed to a single commodity problem. Therefore, it is an interesting open question whether the FTNPB is NP-hard or not on an arbitrary network. Similarly, Myung and Moon (2014) showed that the seemingly two commodity model of relocating both standard and foldable containers by Moon et al. (2013) can be presented as a single commodity model. Both transformations by Myung (2017) and Myung and Moon (2014) commonly used an observation that a constraint for describing the numbers of flows that are bundled and unbundled could be represented as a flow conservation constraint. However, in addition to that observation, the following additional features of each model enable the two models to be transformed to corresponding single commodity network models. Due to the simple star structure considered, Myung (2017) could predetermine the optimal flows of one commodity, and Myung and Moon (2014) simplified the bundling decision via the assumption that all foldable containers should be folded (bundled) when transporting. Therefore, the techniques used Corresponding author. E-mail address: [email protected] (Y.-M. Yu). 1 The present research was conducted by the research fund of Dankook University in 2018. ⁎
https://doi.org/10.1016/j.tre.2019.101827 Received 2 April 2019; Received in revised form 28 November 2019; Accepted 11 December 2019 1366-5545/ © 2019 Elsevier Ltd. All rights reserved.
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by Myung (2017) and Myung and Moon (2014) are not directly applicable to a single commodity formulation of the FTNPB on an arbitrary network. The contributions of this paper can be summarized as follows. First, we analyze the computational complexity of the FTNPB on an arbitrary network and prove that the problem is NP-hard. Secondly, because solving large-scale NP-hard problems is very time consuming, we develop an efficient heuristic to obtain near-optimal solutions within reasonable time. The key feature of our heuristic is that it uses a network flow algorithm. We also conduct intensive computing experiments to test our algorithm. Thirdly, the standard FTNPB assumes that a fixed number of flows are allowed to be bundled together, e.g., Shintani et al. (2010) used exactly four. We consider the variable bundling case where arbitrary numbers of flows can be bundled up to a given limit. We also show that the variable bundling version is still NP-hard and that a slight modification of our heuristic is directly applicable to this case. The remainder of the paper is organized as follows. In Section 2, we discuss some known and possible applications of the FTNPB and related studies. We formally describe the FTNPB and present an integer programming formulation of the problem in Section 3. Section 4 investigates the complexity of the problem, and in Section 5, we present a network flow algorithm-based heuristic to solve the proposed integer programming model. In Section 6, we consider the variable bundling version. We report the computational results in Section 7, and concluding remarks are presented in Section 8. 2. Applications and related studies In this section, we explain the application of the FTNPB in the empty foldable container relocation planning introduced by Shintani et al. (2010) and consider some possible applications in multimodal freight transportation. We also consider related studies. 2.1. Empty foldable container relocation Container transportation is a key component in current logistics, and allocating containers to meet demand is a challenging problem for a shipping company. As the demand and supply of empty containers are usually imbalanced at customer sites, a shipping company must reposition empty containers and consider the use of foldable containers to reduce transportation cost for repositioning. Recently, foldable containers have been produced by Holland Container Innovation, Korea Railway Research Institute, and various other companies. Usually, either 4 or 6 foldable containers can be folded and bundled and occupy the space of a single regular container. Although foldable containers reduce transportation cost, their usage also involves folding and unfolding costs. Therefore, the company must carefully consider whether using foldable containers to reduce the total cost of empty container relocation. After Konings and Thijs (2001) and Konings (2005) analyzed the potential cost savings of foldable containers, a variety of transportation models using foldable containers have been studied. Shintani et al. (2012), Moon et al. (2013), Myung and Moon (2014), and Moon and Hong (2016) considered multi-period container planning models for ocean transportation. These models include more complex decisions in addition to foldable container repositioning, such as purchasing and stocking containers. Zazgornik et al. (2012) proposed a combined model of vehicle routing and foldable container scheduling. Shintani et al. (2010) and Myung (2017) dealt with the problem of relocating empty containers in hinterland transportation. Foldable container reposition was combined with ship type decision-making by Wang et al. (2017), incorporated into drayage services by Zhang et al. (2018a), and into the intermodal transportation network of the Belt and Road Initiative by Zhang et al. (2018b). Other empty container allocation problems without considering foldable containers have been studied by Crainic et al. (1993), Shen and Khoong (1995), Cheang and Lim (2005), Li et al. (2007), Shintani et al. (2007), Dong and Song (2009), Song and Carter (2009), Moon et al. (2010), Song and Dong (2010), Song and Zhang (2010), Meng and Wang (2011), and Song and Dong (2011). As folding and unfolding can be represented as bundling and unbundling, the FTNPB is applicable to foldable container relocation. In fact, the models for foldable containers introduced by both Shintani et al. (2010) and Myung (2017) are equivalent to the FTNPB. Shintani et al. (2010) developed an integer programming formulation, and Myung (2017) showed that when the underlying network has star structure, the model can be solved using a network flow algorithm, i.e., solved in polynomial time. The FTNPB does not seem to be directly applicable to the other models with complex decisions combined with foldable container repositioning. However, the FTNPB is imbedded in those models and its properties can be used when analyzing the substructures of the models. 2.2. Multimodal freight transportation In this subsection, we consider possible applications of the FTNPB in multimodal freight transportation. We consider these applications in multimodal transportation because bundling/unbundling can represent the operations required for switching transportation modes. The scope of multimodal freight transportation is very broad, and a myriad of research has dealt with various multimodal freight transportation problems. Good surveys of multimodal freight transportation were conducted by Crainic and Kim (2007) and SteadieSeifi et al. (2014). The FTNPB can be applied to tactical and/or operational planning. The FTNPB is a single commodity problem and has a particular assumption that the transportation cost of flow depends not only on its amount but also whether it is bundled or not. However, most multimodal problems are modelled on multicommodity networks, and no models equivalent or similar to the FTNPB have been found in the literature. Here, we introduce a possible application of the FTNPB. In Korea, most land transportation companies provide shipment services using two types of trucks: one-ton trucks and four-ton trucks. Companies use one-ton trucks for the pickup process and four-ton trucks for long-distance transportation. To save transportation cost, companies can use both trucks in a transport chain by moving the cargos loaded in four one-ton trucks into one four-ton truck. This freight consolidation is a good option, especially in Korea, because four-ton 2
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trucks are restricted from passing through most city areas. The operation costs for the two types of trucks are similar, but loading and unloading incur costs. Therefore, Korean land transportation companies can obtain an optimal decision for their multimodal freight transportation by solving the FTNPB. Besides the above special instance, we can consider a more general situation where the FTNPB can be applied. Suppose that a company has multiple plants and multiple warehouses and a railway station exists near each plant and warehouse. The company can ship goods from a plant to a warehouse not only by truck but also by a truck-rail-truck multimodal chain. When the latter multimodal chain is chosen, goods are shipped from a plant to a nearby station by truck, moved by train between the two stations, and finally moved from the arrival station to a warehouse by truck. If we assume that the transshipment between truck and train incurs cost and that the transportation cost of the train is charged as a flat fee per cargo bay, which carries 3 or 4 times the amount of a unit truckload. Then, the FTNPB models the problem of determining minimum cost shipment of goods that satisfies the demands at warehouses from available supplies at plants. 3. Problem description and mathematical formulation In this section, we formally state the FTNPB and develop an integer programming formulation of the problem. Consider a directed network G = (N , A) that represents a transportation network. Each node in N is either a supply node or a demand node. Each arc corresponds to a transportation service and has an associated transportation cost. Let ND be the set of demand nodes and NS be the set of supply nodes, i.e., N = NS ND . Transshipment nodes, if they exist, are classified as supply nodes with zero supplies. Let Di be the demand of demand node i and Si be the supply of supply node i . The FTNPB aims to determine how to send flows from supply nodes to satisfy the demands of demand nodes at minimum cost. Unlike an ordinary single commodity network flow problem, we can send multiple flows, after grouping into a bundle, at the cost of one unit flow. We assume that no flows are given in a bundle at all supply nodes, and if the flows arrive in a bundle at demand nodes, they must be separated, i.e., unbundled. The bundling and unbundling also incur costs. Our objective is to minimize the total cost, which consists of transporting, bundling, and unbundling costs. We use the following notations to describe the cost coefficients:
Cij : unit transportation cost from node i to node j CiB : unit bundling cost at node i CiU : unit unbundling cost at node i The distinct characteristics of the FTNPB is that a certain number of flows can be grouped in a bundle and the transportation cost of one bundle on each arc is the same as that of a unit flow. We assume that b flows are grouped in a single bundle and thus the 1 transportation cost of a unit flow in a bundle from node i to node j is b Cij . We assume that the transportation cost of one bundle equal to that of a unit flow as in Shintani et al. (2010) and Myung (2017) but even in case of arbitrary transportation cost per bundle, the whole results of this study would not make any difference. We also assume that exactly b flows are grouped in a bundle as in Shintani et al. (2010) and Myung (2017). This restriction can be relaxed and we will consider the case where arbitrary number of flows are allowed to be bundled in Section 6. We assume that all the parameters are given as integers and that i NS Si = i ND Di . Now we describe the problem in mathematical form and use the following notations as decision variables:
x ij : amount of flows transported not in a bundle from node i to node j yij : number of bundles transported from node i to node j z iB : number of flows bundled at node i z iU : number of flows unbundled at node i Then, the total cost can be expressed as (i, j) A Cij (x ij + yij ) + i N (CiB z iB + CiU z iU ) , where the flows transported from node i to node j are represented by x ij + byij . Notice that at each node, if there are more outgoing bundles than incoming bundles, the bundling operation would be done, and if the opposite case occurs, the unbundling operation would be done. Therefore, z iB and z iU can be described as follows:
z iB = max b
yij {j (i, j) A}
z iU = max b
yji , 0 , { j (j , i ) A }
yji {j (j , i) A}
yij , 0 {j (i, j ) A}
Now, the integer programming formulation for the FTNPB is as follows:
(P)minimize
(CiB z iB + CiU z iU )
Cij (x ij + yij ) + (i, j ) A
i N
3
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subject to
(x ij + byij )
(xji + byji ) =
{j (i, j) A}
b
{ j (j , i ) A }
yji = z iB
yij {j (i, j ) A}
z iU
i
Si, if i NS Di , if i ND
(1)
N
(2)
{j (j, i) A}
x ij , yij , z iB , z iU
0 and integer
(i , j )
A, i
N
(3)
The constraints in (1) ensure flow conservation at each node. The constraints in (2) describe the relations among the variables. yij , z iB , and z iU have already been described. 4. Problem complexity In this section, we study the computational complexity of (P). As we already mentioned, Myung (2017) proved that if a given network is a star network, (P) can be solved in polynomial time. Here, we show that (P) is NP-hard on an arbitrary network. For this purpose, we introduce the following NP-complete decision version of 3-EXACT COVER (X3C): X3C
INSTANCE: A family F = {S1, , Sn} of three element subsets of S = {a1, QUESTION: Is there a subfamily of m subsets that covers S ?
, a3m} .
Theorem 1. (P) is NP-hard. Proof. We show that X3C is reducible to the decision version of (P). Given an instance of X3C, an instance of (P) is associated with it by the following procedure. We construct a transportation network that has 3m + n supply nodes and one demand node such that NS = NS1 NS2 , where NS1 = {1, , 3m} , NS2 = {3m + 1, , 3m + n} , and ND = {3m + n + 1} . Set Si = 1 for all i NS1, Si = 0 for all i NS2 , and D3m + n + 1 = 3m . We also establish the set of arcs in the network such that A = A1 A2 , where A1 = {(i, 3m + j) i NS1, 3m + j NS2, and ai Sj} and A2 = {(3m + i, 3m + n + 1) 3m + i NS2}. We assume that three flows are grouped in a bundle, i.e., b = 3. Set Cij = 0 for all (i, j) A1 and Cij = 1 for all (i, j) A2 . We also set CiB = CiU = 0 for all i N . Then, the answer to the X3C instance is yes if and only if the associated (P) has a feasible solution with cost m. □ 5. Solution method In this section, we develop a solution for the integer programming problem (P) presented in Section 2. As (P) is NP-hard, it would take huge CPU time to solve large-sized problems exactly. This was already observed by Shintani et al. (2010) and was proven to be true through our computing experiments that we present later. For this reason, we develop a heuristic to obtain near-optimal solutions within reasonable time. Our heuristic is based on the linear programming relaxation of (P), denoted by (LP), where the integrality condition in (3) is replaced with the non-negativity condition. Our algorithm first solves (LP) and fixes the values of the variables yij using the optimal solution of (LP). Then, we determine variables z iB and z iU through the constraints in (2) and the remaining variables x ij by solving the following subproblem (SP) of (P):
(SP) minimize
Cij xij (i, j) A
subject to
x ij {j (i, j ) A}
x ij
0 and integer
Si, if i
xji =
D i , if i
{j (j, i) A}
(i , j )
NS (4)
ND
(5)
A
{
}
{
}
byij byji and Di = Di + byij byji . where Si = Si {j (i, j) A} { j (j , i ) A } {j (i, j ) A} {j (j , i) A} The main concept behind our heuristic is based on the observation that both (LP) and (SP) can be viewed as network flow problems. It is straightforward that (SP) is a network flow problem because the constraints in (4) are the flow conservation constraints. However, it is non-trivial to show that we can solve (LP) using a network flow algorithm. We formally verify our findings as follows: Lemma 1. (LP) can be transformed to a network flow problem. Proof. The constraints in (1) can be modified to the following constraint using (2):
x ji + z iB
x ij {j (i, j) A}
{j (j, i) A}
ziU =
Si, if i NS Di , if i ND
(1′)
Replacing the term byij with fij , (LP) can be transformed into the following equivalent problem: 4
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Fig. 1. Decision variables corresponding to arcs.
(LP') minimize
Cij x ij + (i, j ) A
subject to
x ij , fij
z iU =
{j (j, i) A}
f ji = z iB
fij
(CiB z iB + CiU ziU ) i N
xji + z iB
x ij {j (i, j) A}
{j (i, j) A}
1 f + b ij
z iU
i
Si, if i NS Di , if i ND
N
(2′)
{ j (j , i ) A }
0
(i , j )
(1′)
A
Now we show that (LP') is a network flow problem. The constraints in (1') and (2') can be seen as the flow conservation constraints in a network. The corresponding network is shown in Fig. 1, where node i and node i' correspond to the constraints in (1') and (2'), respectively, and each arc corresponds to the variable labeled on the arc. □ As (LP) and (SP) are network flow problems, both problems can be solved using an efficient network flow algorithm instead of a general-purpose linear and/or integer programming method. More details on network flow problems and algorithms are given by Ahuja et al. (1993). As we mentioned, our algorithm first solves (LP') and determines the values of yij using the optimal solution of (LP'). An optimal solution of (LP') satisfies integrality and is also optimal to (P) if the optimal values of variables fij are multiples of b. In that case, yij = 1 bfij for each (i, j) A, and the optimal values of the remaining variables x ij , z iB , and z iU from (LP') are also optimal for (P). On the other hand, if any fij is not a multiple of b, we set yij = 1 bfij , determine the variables z iB and z iU through the constraints in (2), and solve (SP) to obtain the values of the variables x ij . 6. Variable bundling Up to now, we have assumed that exactly b flows are grouped in a bundle. However, if we allow arbitrary numbers of flows to be bundled, the resulting FTNPB would be more versatile and could be used in more applications. In this section, we consider the extended model, which we call the FTNPB with variable bundling, where up to b flows can be bundled. We analyzed the complexity of the extended model and show that our heuristic is directly applicable to the problem. 6.1. Problem complexity The FTNPB with variable bundling is still NP-hard. To prove this, we can use the proof of Theorem 1. Consider the same transportation network in the proof; the X3C instance is yes if and only if the FTNPB with variable bundling has a feasible solution with cost m. 6.2. Solution method First, we develop the mathematical programming formulation of the variable bundling version. As we can bundle 2, 3, , or b flows, we require the variables yijk for 2 k b instead of yij to represent the number of bundles grouping k flows. Then, the new problem can be formulated as follows:
Cij (xij + yij2 +
(PV) minimize (i, j) A
+ yijb ) +
(CiB z iB + CiU ziU ) i N
5
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Y.-S. Myung and Y.-M. Yu
subject to
{j (i, j ) A}
(x ij + 2yij2 +
+ byijb )
(2yji2 +
{j (i, j) A}
{j (j, i) A}
(xji + 2yji2 +
+byjib )
Si, if i NS Di , if i ND
=
(2yij2 +
+ byijb )
(6)
+ byjib ) = z iB
ziU
i
N
(7)
{ j (j , i ) A }
x ij , yijk , ziB , ziU
0 and integer
(i , j )
A, i
N, 2
k
b
(8)
We show that our heuristic developed in Section 5 can solve (PV) by a slight modification. Like the heuristic for (P), our new heuristic solves the linear programming relaxation of (PV), denoted by (LPV), determines yijk with 2 k b by truncating the
fractional solution of (LPV), sets z iB and z iU through the constraints in (7), and solves the following subproblem (SPV) to find x ij :
(SPV) minimize
Cij xij (i, j) A
subject to
x ij {j (i, j ) A}
x ij
D i , if i
{j (j, i) A}
0 and integer
{
Si, if i
xji = (i , j )
NS (9)
ND
(10)
A
}
{
}
+byjib ) . where Si = Si and Di = Di + (SPV) is exactly the same as (SP) and is a network flow problem. All that remains is to show that (LPV) can be solved using a network flow algorithm as for (LP). This can be explained in the same way as the proof of Lemma 1. We restate it as follows:
(2yij2 + {j (i, j) A}
+ byijb )
(2yji2 + {j (j, i) A}
+ byjib )
(2yij2 + {j (i, j) A}
+byijb )
(2yji2 + {j (j, i ) A}
Lemma 2. (LPV) can be transformed to a network flow problem. Proof. The constraints in (6) can be modified to the following constraint using (7):
x ji + z iB
x ij {j (i, j) A}
Replacing
{j (j, i) A}
kyijk
with
fijk
for 2
(LPV') minimize
Cij x ij +
subject to
x ij , fijk
(fij2 +
0
+ fijb )
(i , j )
1 2 f + 2 ij
1 + fijb + b
xji + z iB
x ij {j (i, j) A}
z iU =
{j (j, i) A}
{j (j, i) A}
A, 2
(6′)
b , (LVP) can be transformed into the following equivalent problem:
k
(i, j ) A
{j (i, j) A}
Si, if i NS Di , if i ND
ziU =
(f ji2 +
k
+f jib ) = z iB
(CiB z iB + CiU z iU ) i N
Si, if i NS Di , if i ND z iU
i
N
(6′) (7′)
b
As in (LP'), the constraints in (6' ) and (7') can be seen as the flow conservation constraints in a network. The corresponding network is the same as the network shown in Fig. 1 except that there are multiple edges from node i to node j , each of which corresponds to fijk . □ 7. Computational results In this section, we describe and report the results of computing experiments to evaluate the performance of our heuristic and to analyse the effects of bundling. We also conduct the sensitivity analysis to find how the changes of cost parameters affect the total cost. 7.1. Performance of our heuristic For intensive tests, we randomly generated data instances with various sizes. We first drew a 100 × 100 rectangle on which node sites were randomly located, and networks were generated with various numbers of nodes. Arcs were also randomly selected until the resulting network density (ratio of generated arcs to total number of possible arcs) reached the predetermined level (20, 35, or 50%). We generated 5 instances for each combination of a number of nodes and network density. The cost parameters for transportation, bundling, and unbundling were set as in Shintani et al. (2010). We assumed that a fixed number of flows was allowed to be bundled and set b = 4 . Test runs were performed on a PC (2.4 GHz and 8G RAM). 6
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Table 1 Computational results for small-sized problems. No. of nodes
10
Network density
0.2
0.35
0.5
15
0.2
0.35
0.5
20
0.2
0.35
0.5
z(H)
z(P)
z(LP)
z (H ) z (P )
z (H ) z (LP )
z (P ) z (LP )
Computation time (seconds) Heuristic
Optimal
23693.0 27954.0 21511.0 25395.0 22483.0 20452.0 29653.0 20150.0 14941.0 23786.0 17401.0 23034.0 14480.0 23093.0 21069.0
23398.0 27615.0 21434.0 25252.0 22483.0 20452.0 29197.0 18812.0 13807.0 22898.0 15784.0 21673.0 13543.0 22244.0 20283.0
21790.8 26082.0 20552.8 23162.5 21183.2 19540.2 28106.5 17758.8 12683.8 21584.0 14868.0 20900.5 12403.5 21400.8 19525.3
1.01 1.01 1.00 1.01 1.00 1.00 1.02 1.07 1.08 1.04 1.10 1.06 1.07 1.04 1.04
1.09 1.07 1.05 1.10 1.06 1.05 1.06 1.13 1.18 1.10 1.17 1.10 1.17 1.08 1.08
1.07 1.06 1.04 1.09 1.06 1.05 1.04 1.06 1.09 1.06 1.06 1.04 1.09 1.04 1.04
0.031 0.016 0.015 0.016 0.031 0.016 0.016 0.015 0.016 0.015 0.016 0.015 0.031 0.016 0.031
0.047 0.047 0.016 0.063 0.203 0.094 0.110 0.016 0.062 0.062 0.047 0.047 0.156 0.047 0.047
29278.0 32269.0 29806.0 30586.0 25319.0 34158.0 29161.0 19041.0 25041.0 21648.0 25182.0 21649.0 30797.0 22810.0 28455.0
28297.0 31582.0 28018.0 30513.0 24886.0 33163.0 27777.0 17447.0 24624.0 20944.0 22937.0 20585.0 29336.0 21582.0 27076.0
26422.0 30405.8 26207.5 28675.5 23026.5 31568.8 26724.0 16315.0 23246.8 19935.8 21737.3 19460.3 28192.8 20363.8 25745.8
1.03 1.02 1.06 1.00 1.02 1.03 1.05 1.09 1.02 1.03 1.10 1.05 1.05 1.06 1.05
1.11 1.06 1.14 1.07 1.10 1.08 1.09 1.17 1.08 1.09 1.16 1.11 1.09 1.12 1.11
1.07 1.04 1.07 1.06 1.08 1.05 1.04 1.07 1.06 1.05 1.06 1.06 1.04 1.06 1.05
0.031 0.031 0.015 0.031 0.016 0.016 0.016 0.015 0.015 0.031 0.031 0.015 0.016 0.031 0.031
0.063 0.031 0.063 0.063 0.047 0.141 0.062 0.047 0.093 0.047 0.047 0.078 0.219 0.031 0.031
31500.0 33834.0 30400.0 33577.0 37158.0 29720.0 26371.0 32871.0 31585.0 29310.0 32175.0 38664.0 27187.0 38727.0 33645.0
29529.0 31783.0 30013.0 31426.0 35072.0 28106.0 25331.0 31547.0 30163.0 26371.0 31506.0 37001.0 26407.0 37537.0 32049.0
27712.5 29559.0 28324.8 29644.8 32597.5 26481.3 23676.3 30068.5 28623.3 25052.0 29975.0 35383.3 25190.3 36278.8 30583.5
1.07 1.06 1.01 1.07 1.06 1.06 1.04 1.04 1.05 1.11 1.02 1.04 1.03 1.03 1.05
1.14 1.14 1.07 1.13 1.14 1.12 1.11 1.09 1.10 1.17 1.07 1.09 1.08 1.07 1.10
1.07 1.08 1.06 1.06 1.08 1.06 1.07 1.05 1.05 1.05 1.05 1.05 1.05 1.03 1.05
0.031 0.015 0.015 0.015 0.031 0.031 0.015 0.031 0.031 0.015 0.015 0.016 0.031 0.016 0.063
0.187 0.078 0.218 0.031 0.047 0.156 0.031 0.062 0.140 0.093 0.062 0.063 0.235 0.109 0.047
We classified the data instances into three groups. The first group included relatively small-sized data instances with 10, 15, and 20 nodes; the computational results for this group are summarized in Table 1. The second group consisted of medium-sized problems with 30, 50, and 100 nodes (Table 2) and the last group was large-sized problems with 150, 200, and 300 nodes (Table 3). For each group of problems, we evaluated the performance of our algorithm by comparing the objective value of our heuristic (z(H)) with the objective value obtained from (P) (z(P)). We set a time limit of 100 s, and we could not exactly solve (P) for several large-sized problems within the time limit. Therefore, we also provide the objective value obtained from (LP) (z(LP)), which is a lower bound of z (P). In each table, we report the computation times for our heuristic and an integer programming algorithm. When solving (LP), we used a general-purpose linear programming solver instead of a network flow algorithm for computational convenience. Nevertheless, the computational advantages of our heuristic over an integer programming algorithm are clearly reflected in the results. We used optimization software CPLEX 12.9 to solve both an integer and linear programming problem. The results revealed that our heuristic finds near-optimal solutions for fairly large problems within reasonable time. Note that in all instances, the linear programming relaxation does not provide an integer solution but provides very good ratios of z(LP) to z(P). This good integrality gap seems to be due to the network structure of the FTNPB. For each combination of number of nodes and network density, most of the average ratios of z(H) to z(P) are within 5%, and the ratios of z(H) to z(LP) are within 10%. Also, note that the computation times of our heuristic to solve the large-sized problems in Table 3 are within a couple of seconds, while the 7
Transportation Research Part E 133 (2020) 101827
Y.-S. Myung and Y.-M. Yu
Table 2 Computational results for medium-sized problems. No. of nodes
30
Network density
z(H)
0.2
42465.0 53426.0 48169.0 47581.0 55823.0 43072.0 44468.0 39998.0 53901.0 60008.0 47188.0 53264.0 42098.0 39177.0 44825.0
40673.0 49817.0 46092.0 46328.0 55287.0 41166.0 41095.0 37540.0 52023.0 57460.0 44698.0 50945.0 40302.0 36906.0 42368.0
38488.3 47752.5 43460.5 43521.3 52766.0 38639.0 38761.3 35816.3 49998.5 55548.3 42963.5 48984.3 38391.8 35008.3 40514.0
1.04 1.07 1.05 1.03 1.01 1.05 1.08 1.07 1.04 1.04 1.06 1.05 1.04 1.06 1.06
1.10 1.12 1.11 1.09 1.06 1.11 1.15 1.12 1.08 1.08 1.10 1.09 1.10 1.12 1.11
72063.0 90595.0 78435.0 81737.0 69791.0 78205.0 81875.0 64877.0 71617.0 82246.0 73350.0 68071.0 71409.0 64988.0 68726.0
69412.0 86847.0 74728.0 77399.0 65696.0 75929.0 79231.0 61896.0 68892.0 79842.0 69494.0 64465.0 67909.0 63140.0 64445.0
65570.5 83290.3 70823.5 73287.8 62102.0 72372.3 76404.0 58964.0 65950.8 76546.8 66641.8 61683.8 64940.0 60386.8 61348.0
1.04 1.04 1.05 1.06 1.06 1.03 1.03 1.05 1.04 1.03 1.06 1.06 1.05 1.03 1.07
163470.0 152857.0 159611.0 135936.0 133749.0 147771.0 133777.0 131094.0 142768.0 137010.0 157365.0 136342.0 147947.0 130119.0 155289.0
154228.0 145578.0 151723.0 129649.0 126214.0 140409.0 128661.0 123937.0 136850.0 130688.0 151256.0* 128713.0 141775.0 123922.0 145895.0
146766.0 139241.3 145262.0 123445.3 119112.5 134255.0 122353.3 118400.0 130717.8 124864.8 146525.3 122727.0 136273.3 117847.3 140249.3
1.06 1.05 1.05 1.05 1.06 1.05 1.04 1.06 1.04 1.05 1.04 1.06 1.04 1.05 1.06
0.35
0.5
50
0.2
0.35
0.5
100
0.2
0.35
0.5
z(P)
z(LP)
z (H ) z (P )
z (H ) z (LP )
z (P ) z (LP )
Computation time (seconds) Heuristic
Optimal
1.06 1.04 1.06 1.06 1.05 1.07 1.06 1.05 1.04 1.03 1.04 1.04 1.05 1.05 1.05
0.047 0.015 0.016 0.032 0.031 0.031 0.031 0.047 0.031 0.047 0.047 0.047 0.047 0.047 0.047
0.328 0.109 0.063 1.125 0.156 0.078 0.078 0.063 0.047 0.109 0.297 0.094 0.125 0.079 0.062
1.10 1.09 1.11 1.12 1.12 1.08 1.07 1.10 1.09 1.07 1.10 1.10 1.10 1.08 1.12
1.06 1.04 1.06 1.06 1.06 1.05 1.04 1.05 1.04 1.04 1.04 1.05 1.05 1.05 1.05
0.063 0.047 0.047 0.047 0.047 0.062 0.078 0.078 0.078 0.094 0.093 0.094 0.094 0.110 0.093
0.422 0.970 0.328 0.359 0.109 1.343 0.375 0.344 0.156 0.406 0.484 1.547 1.594 0.141 0.219
1.11 1.10 1.10 1.10 1.12 1.10 1.09 1.11 1.09 1.10 1.07 1.11 1.09 1.10 1.11
1.05 1.05 1.04 1.05 1.06 1.05 1.05 1.05 1.05 1.05 1.03 1.05 1.04 1.05 1.04
0.171 0.156 0.172 0.187 0.172 0.250 0.250 0.265 0.266 0.250 0.360 0.359 0.375 0.344 0.359
1.063 1.203 1.172 0.750 0.891 1.110 3.125 1.329 1.281 4.532 > 100 0.828 7.937 0.781 2.828
* Best solution obtained within the time limit.
integer programming solver exceeded the time limit of 100 s for half of the problems. 7.2. Effects of bundling In order to investigate the effects of bundling, we conducted the experiments of comparing the total costs of the three FTNP models: a model with fixed bundling; a model with variable bundling; and a model without bundling. The results given in Table 4 imply that bundling saves the total cost much and that additional cost saving by variable bundling is not that much. 7.3. Sensitivity analysis on cost parameters We also conducted the sensitivity analysis to show how the changes of transportation cost and folding/unfolding cost affect the total cost. The results given in Table 5 revealed that the total cost increases almost at the same rate as transportation cost when bundling is not allowed but increases at lower late when bundling is allowed. In Table 6, the results of the sensitivity analysis on folding/unfolding cost are also provided. 8
Transportation Research Part E 133 (2020) 101827
Y.-S. Myung and Y.-M. Yu
Table 3 Computational results for large-sized problems. No. of nodes
150
Network density
z(H)
0.2
204688.0 207595.0 205149.0 222667.0 230598.0 208490.0 194802.0 200115.0 226844.0 221712.0 215497.0 230988.0 211669.0 203931.0 205639.0
194657.0 197335.0 194895.0 214183.0 221541.0 198671.0 186658.0 192276.0 214968.0 212666.0 205513.0 218942.0 202570.0 193738.0 195133.0
185738.8 187784.5 186010.3 206222.3 212553.3 188931.8 178229.8 184203.8 207092.3 204433.5 196842.3 210081.5 194616.8 185799.8 187501.0
1.05 1.05 1.05 1.04 1.04 1.05 1.04 1.04 1.06 1.04 1.05 1.06 1.04 1.05 1.05
1.10 1.11 1.10 1.08 1.08 1.10 1.09 1.09 1.10 1.08 1.09 1.10 1.09 1.10 1.10
292479.0 291000.0 273744.0 293856.0 280506.0 266088.0 301675.0 250734.0 259594.0 264132.0 265419.0 272746.0 303751.0 247738.0 279311.0
276275.0 277528.0 260047.0 279763.0 267197.0 253199.0 285727.0* 238093.0 245313.0 252248.0 251305.0 260688.0 289935.0 237919.0 265892.0
265297.0 265987.5 248241.0 268203.3 255556.0 241259.0 274043.8 227006.5 233126.3 240845.8 239619.3 249617.5 278264.5 226886.0 254040.0
1.06 1.05 1.05 1.05 1.05 1.05 1.06 1.05 1.06 1.05 1.06 1.05 1.05 1.04 1.05
406073.0 375927.0 381221.0 427440.0 414184.0 376241.0 380873.0 385992.0 414638.0 387404.0 425470.0 417038.0 423032.0 386860.0 393710.0
387399.0 358847.0 361197.0 407240.0 393919.0 359180.0* 365228.0 368624.0* 399088.0 370808.0* 404653.0* 395690.0* 406578.0* 369521.0* 375706.0*
370554.5 341852.5 344188.5 389028.0 376806.5 343185.8 349269.8 350809.0 382294.8 354757.5 388347.8 378426.8 390596.3 353311.0 360564.8
1.05 1.05 1.06 1.05 1.05 1.05 1.04 1.05 1.04 1.04 1.05 1.05 1.04 1.05 1.05
0.35
0.5
200
0.2
0.35
0.5
300
0.2
0.35
0.5
z(P)
z(LP)
z (H ) z (P )
z (H ) z (LP )
z (P ) z (LP )
Computation time (seconds) Heuristic
Optimal
1.05 1.05 1.05 1.04 1.04 1.05 1.05 1.04 1.04 1.04 1.04 1.04 1.04 1.04 1.04
0.391 0.360 0.391 0.359 0.375 0.562 0.563 0.562 0.578 0.578 0.781 0.797 0.797 0.812 0.781
14.016 1.515 1.672 6.750 2.110 10.281 15.453 20.157 14.422 20.281 23.125 16.969 26.890 1.891 23.922
1.10 1.09 1.10 1.10 1.10 1.10 1.10 1.10 1.11 1.10 1.11 1.09 1.09 1.09 1.10
1.04 1.04 1.05 1.04 1.05 1.05 1.04 1.05 1.05 1.05 1.05 1.04 1.04 1.05 1.05
0.594 0.719 0.594 0.594 0.594 0.969 1.047 1.000 1.015 0.984 1.422 1.454 1.484 1.438 1.469
2.938 3.016 17.375 3.282 11.766 28.469 > 100 29.969 32.313 9.781 44.375 61.375 51.484 34.078 52.438
1.10 1.10 1.11 1.10 1.10 1.10 1.09 1.10 1.08 1.09 1.10 1.10 1.08 1.09 1.09
1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.04 1.05 1.04 1.05 1.04 1.05 1.04
1.375 1.343 1.344 1.406 1.375 2.375 2.469 2.500 2.468 2.484 3.781 3.532 3.735 3.641 3.500
62.671 48.141 11.859 19.484 37.046 > 100 58.907 > 100 91.094 > 100 > 100 > 100 > 100 > 100 > 100
* Best solution obtained within the time limit.
8. Conclusions In this paper, we considered the freight transportation network problem with the bundling option. In this model, when we choose the bundling option, we can send multiple goods, after grouping into a bundle, at the cost of one unit good, but there are costs for bundling and unbundling. Our problem is to decide how to send goods with an additional decision for bundling from suppliers to customers at minimum cost. The problem originally arose from relocating empty containers in hinterland transportation, but it can also be applied to multimodal freight transportation. We have shown that the problem is NP-hard and that it is time consuming to solve a large-sized problem exactly using an integer programming algorithm. We found that the problem has a network flow problem structure, and we developed an efficient heuristic that uses a network flow algorithm. We also presented the results of computational experiments and showed that our heuristic finds near-optimal solutions even for fairly large problems within a reasonable time.
9
Transportation Research Part E 133 (2020) 101827
Y.-S. Myung and Y.-M. Yu
Table 4 Cost comparison among different bundling strategies. No. of nodes
Network density
50
No bundling(a)
0.2
150298.0 183545.0 163874.0 187395.0 130080.0 135717.0 147420.0 128992.0 138591.0 151199.0 138827.0 128555.0 131116.0 120379.0 130460.0
0.35
0.5
Fixed bundling (4 flows)
Variable bundling (2 3 4 flows)
(b)
(b/a)
(c)
(c/a)
69412.0 86847.0 74728.0 77399.0 65696.0 75929.0 79231.0 61896.0 68892.0 79842.0 69494.0 64465.0 67909.0 63140.0 64445.0
0.46 0.47 0.46 0.41 0.51 0.56 0.54 0.48 0.50 0.53 0.50 0.50 0.52 0.52 0.49
67540.0 85413.0 73026.0 75214.0 64002.0 74264.0 78199.0 60446.0 67576.0 78273.0 68269.0 63133.0 66480.0 62076.0 62937.0
0.45 0.47 0.45 0.40 0.49 0.55 0.53 0.47 0.49 0.52 0.49 0.49 0.51 0.52 0.48
Table 5 Sensitivity analysis on the increases of transportation cost. (No bundling case) No. of nodes
50
Network density
0.2
0.35
0.5
Original data (a)
150298.0 183545.0 163874.0 187395.0 130080.0 135717.0 147420.0 128992.0 138591.0 151199.0 138827.0 128555.0 131116.0 120379.0 130460.0
10% increase
20% increase
30% increase
40% increase
(b)
(c)
(b)
(c)
(b)
(c)
(b)
(c)
165345.0 201904.0 180349.0 206206.0 143113.0 149388.0 162189.0 142108.0 152476.0 166486.0 152727.0 141499.0 144280.0 132495.0 143627.0
10.0% 10.0% 10.1% 10.0% 10.0% 10.1% 10.0% 10.2% 10.0% 10.1% 10.0% 10.1% 10.0% 10.1% 10.1%
180401.0 220285.0 196750.0 225136.0 156150.0 162969.0 176937.0 154990.0 166362.0 181616.0 166658.0 154364.0 157422.0 144490.0 156642.0
20.0% 20.0% 20.1% 20.1% 20.0% 20.1% 20.0% 20.2% 20.0% 20.1% 20.0% 20.1% 20.1% 20.0% 20.1%
195465.0 238759.0 213238.0 243839.0 169257.0 176622.0 191727.0 167915.0 180252.0 196699.0 180569.0 167274.0 170578.0 156570.0 169740.0
30.1% 30.1% 30.1% 30.1% 30.1% 30.1% 30.1% 30.2% 30.1% 30.1% 30.1% 30.1% 30.1% 30.1% 30.1%
210501.0 257147.0 229605.0 262666.0 182319.0 190180.0 206550.0 180919.0 194202.0 211893.0 194514.0 180248.0 183692.0 168706.0 182942.0
40.1% 40.1% 40.1% 40.2% 40.2% 40.1% 40.1% 40.3% 40.1% 40.1% 40.1% 40.2% 40.1% 40.1% 40.2%
(Fixed bundling case) No. of nodes
50
Network density
0.2
0.35
0.5
(c) =
(b ) (a) (a)
Original data (a)
69412.0 86847.0 74728.0 77399.0 65696.0 75929.0 79231.0 61896.0 68892.0 79842.0 69494.0 64465.0 67909.0 63140.0 64445.0
10% increase
20% increase
30% increase
40% increase
(b)
(c)
(b)
(c)
(b)
(c)
(b)
(c)
63260.0 83219.0 71820.0 74791.0 62808.0 72061.0 75299.0 59324.0 65768.0 76026.0 66382.0 61477.0 64610.0 60024.0 61633.0
6.1% 5.8% 6.2% 6.7% 5.7% 5.2% 5.2% 5.9% 5.6% 5.4% 5.6% 5.6% 5.4% 5.5% 5.7%
58903.0 79599.0 68746.0 71915.0 59780.0 68339.0 71547.0 56602.0 62792.0 72064.0 63186.0 58561.0 61391.0 57180.0 58777.0
12.2% 11.6% 12.3% 13.4% 11.3% 10.3% 10.3% 11.7% 11.2% 10.7% 11.2% 11.2% 10.8% 10.9% 11.4%
54823.0 75503.0 65690.0 69356.0 56848.0 64559.0 67515.0 53842.0 59524.0 68332.0 60040.0 55673.0 58237.0 54122.0 55892.0
18.3% 17.5% 18.4% 20.0% 17.0% 15.5% 15.5% 17.4% 16.9% 16.0% 16.9% 16.8% 16.2% 16.4% 17.2%
51087.0 71883.0 62970.0 66752.0 53912.0 60715.0 63423.0 51066.0 56492.0 64324.0 56800.0 52657.0 54957.0 51242.0 53036.0
24.4% 23.3% 24.6% 26.7% 22.6% 20.6% 20.7% 23.2% 22.5% 21.3% 22.5% 22.4% 21.5% 21.8% 22.9%
× 100(%) .
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Transportation Research Part E 133 (2020) 101827
Y.-S. Myung and Y.-M. Yu
Table 6 Sensitivity analysis on the decreases of folding and unfolding cost. No. of nodes
50
Network density
0.2
0.35
0.5
(c) =
(a) (b ) (a)
Original data (a)
69412.0 86847.0 74728.0 77399.0 65696.0 75929.0 79231.0 61896.0 68892.0 79842.0 69494.0 64465.0 67909.0 63140.0 64445.0
10% decrease
20% decrease
30% decrease
40% decrease
(b)
(c)
(b)
(c)
(b)
(c)
(b)
(c)
63260.0 83219.0 71820.0 74791.0 62808.0 72061.0 75299.0 59324.0 65768.0 76026.0 66382.0 61477.0 64610.0 60024.0 61633.0
8.9% 4.2% 3.9% 3.4% 4.4% 5.1% 5.0% 4.2% 4.5% 4.8% 4.5% 4.6% 4.9% 4.9% 4.4%
58903.0 79599.0 68746.0 71915.0 59780.0 68339.0 71547.0 56602.0 62792.0 72064.0 63186.0 58561.0 61391.0 57180.0 58777.0
15.1% 8.3% 8.0% 7.1% 9.0% 10.0% 9.7% 8.6% 8.9% 9.7% 9.1% 9.2% 9.6% 9.4% 8.8%
54823.0 75503.0 65690.0 69356.0 56848.0 64559.0 67515.0 53842.0 59524.0 68332.0 60040.0 55673.0 58237.0 54122.0 55892.0
21.0% 13.1% 12.1% 10.4% 13.5% 15.0% 14.8% 13.0% 13.6% 14.4% 13.6% 13.6% 14.2% 14.3% 13.3%
51087.0 71883.0 62970.0 66752.0 53912.0 60715.0 63423.0 51066.0 56492.0 64324.0 56800.0 52657.0 54957.0 51242.0 53036.0
26.4% 17.2% 15.7% 13.8% 17.9% 20.0% 20.0% 17.5% 18.0% 19.4% 18.3% 18.3% 19.1% 18.8% 17.7%
× 100(%) .
CRediT authorship contribution statement Young-Soo Myung: Conceptualization, Methodology, Writing - original draft. Yung-Mok Yu: Supervision, Data curation, Software, Writing - review & editing. References Ahuja, R.K., Magnanti, T.L., Orlin, J.B., 1993. Network Flows, Prentice-Hall, New Jersey, USA. Cheang, B., Lim, A., 2005. A network flow bases method for the distribution of empty containers. Int. J. Comput. Appl. Technol. 22, 198–204. Crainic, T.G., Gendreau, M., Dejax, P., 1993. Dynamic and stochastic models for the allocation of empty containers. Oper. Res. 41, 102–126. Crainic, T.G., Kim, K., 2007. Intermodal transportation. Handbooks Oper. Res. Manage. Sci. 14, 467–537. Dong, J.X., Song, D.P., 2009. Container fleet sizing and empty repositioning in liner shipping systems. Transp. Res. Part E 45, 860–877. Fortune, S., Hopcroft, J., Wyllie, J., 1980. The directed subgraph homeomorphism problem. Theoret. Comput. Sci. 10, 111–121. Konings, R., Thijs, R., 2001. Foldable containers: a new perspective on reducing container-repositioning costs. Eur. J. Oper. Res. 1 (4), 333–352. Konings, R., 2005. Foldable containers to reduce the costs of empty transport? A cost–benefit analysis from a chain and multi-actor perspective. Marit. Econ. Logist. 7 (3), 223–249. Li, J.A., Leung, S.C.H., Wu, Y., Liu, K., 2007. Allocation of empty containers between multi-ports. Eur. J. Oper. Res. 182, 400–412. Meng, Q., Wang, L., 2011. Liner shipping service network design with empty container repositioning. Transp. Res. Part E 47, 605–708. Moon, I.K., Do Ngoc, A.D., Hur, Y.S., 2010. Positioning empty containers among multiple ports with leasing and purchasing consideration. OR Spectrum 32, 765–786. Moon, I.K., Do Ngoc, A.D., Konings, R., 2013. Foldable and standard containers in empty container repositioning. Transp. Res. Part E 49, 107–124. Moon, I.K., Hong, H.J., 2016. Repositioning of empty containers using both standard and foldable containers. Marit. Econ. Logist. 18 (1), 61–77. Myung, Y.S., Moon, I.K., 2014. A network flow model for the optimal allocation of both foldable and standard containers. Oper. Res. Lett. 42, 484–488. Myung, Y.S., 2017. Efficient solution methods for the integer programming models of relocating empty containers in the hinterland transportation network. Transp. Res. Part E 108, 52–59. Shen, W.S., Khoong, C.M., 1995. A DSS for empty container distribution planning. Decis. Support Syst. 15, 75–82. Shintani, K., Konings, R., Imai, A., 2010. The impact of foldable containers on container fleet management costs in hinterland transport. Transp. Res. Part E 46, 750–763. Shintani, K., Konings, R., Imai, A., 2012. The impact of foldable containers on the costs of container fleet management in liner shipping networks. Marit. Econ. Logist. 14, 455–479. Shintani, K., Imai, A., Nishimura, E., 2007. The container shipping network design problem with empty container repositioning. Transp. Res. Part E 43, 39–59. Song, D.P., Carter, J., 2009. Empty container repositioning in liner shipping. Marit. Policy Manage. 36, 291–307. Song, D.P., Dong, J.X., 2010. Effectiveness of an empty container repositioning policy with flexible destination ports. Transp. Pol. 18, 92–101. Song, D.P., Dong, J.X., 2011. Flow balancing-based empty container repositioning in typical shipping service routes. Marit. Econ. Logist. 13, 61–77. Song, D.P., Zhang, Q., 2010. A fluid flow model for empty container repositioning policy with a single port and stochastic demand. Siam J. Control Optim. 48 (5), 3623–3642. SteadieSeifi, M., Dellaert, N.P., Nuijten, W., Woensel, T.V., Raoufi, R., 2014. Multimodal freight transportation planning: a literature review. Eur. J. Oper. Res. 233, 1–15. Wang, K., Wang, S., Zhen, L., Qu, X., 2017. Ship type decision considering empty container repositioning and foldable containers. Transp. Res. Part E 108, 97–121. Zazgornik, J., Gronalt, M., Hirsch, P., 2012. The combined vehicle routing and foldable container scheduling problem: a model formulation and tabu search bases solution approaches. INFOR 30, 147–162. Zhang, R., Zhao, H., Moon, I.K., 2018a. Range-based truck-state transition modeling method for foldable container drayage services. Transp. Res. Part E 118, 225–239. Zhang, S., Ruan, X., Xia, Y., Feng, X., 2018b. Foldable container in empty container repositioning in intermodal transportation network of Belt and Road Initiative: strengths and limitations. Maritime Poli. Manage. 45, 351–369.
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