- Email: [email protected]
Two-stage approach to the intermodal terminal location problem
Computers & Operations Research 67 (2016) 113–119
Contents lists available at ScienceDirect
Computers & Operations Research journal homepage: www.elsevier.com/locate/caor
Two-stage approach to the intermodal terminal location problem Chang-Chun Lin, Shih-Wei Lin n Department of Information Management, Chang Gung University, 259 Wen-Hwa 1st Road, Kwei-Shan, Tao-Yuan 333, Taiwan, ROC
art ic l e i nf o
Keywords: Multimodal transportation Intermodal transportation Intermodal terminals 0–1 Programming Matheuristics
a b s t r a c t Multimodal transportation means to transport freight using at least two transportation modes. Intermodal freight transportation, a particular form of multimodal transportation, transports freight in an intermodal container without handling of the freight itself when changing modes. The intermodal terminal location problem involves selecting terminals that constitute an intermodal transportation network and routing freight flows with minimal total transportation and operating costs. Arnold et al. first presented mathematical programming models for the problem. Sörensen et al. recently proposed another model for the problem. However, Lin et al. have shown that the model of Sörensen et al. is complex and time-consuming and therefore developed a modified mixed integer programming model to increase computation efficiency. This study shows that the model of Lin et al. can be further improved by separating the selection of intermodal terminals from the routing of transportation flows. A two-stage programming approach is proposed along with a modified, more efficient matheuristic. & 2015 Elsevier Ltd. All rights reserved.
1. Introduction As global trade expands, long-haul transportation is increasingly important. Unimodal long-haul transportation that uses only one transportation mode (e.g., road, rail, or water) is literally impossible for global trade. Managing multimodal transportation is thus important both practically and academically. Multimodal transportation transports goods by using at least two transportation modes. Intermodal transportation, a particular form multimodal transportation, has received considerable attention in recent years. According to the European Conference of Ministers of Transport [1], intermodal transportation refers to the movement of goods in the same loading unit without handling the goods themselves during transfers between modes. Intermodal transportation is well recognized as environmentally friendly, capable of reducing congestion, accessible, and highly feasible for global trading. The multi-faceted literature on multimodal and intermodal transportation addresses issues such as decision of the number and location of terminals, route selection [2], cost analysis [2,3], transport policy [4] and intermodal decision support [5]. Literature reviews on multimodal transportation followed a common framework of the decision horizon of planning: strategic, tactical, and operational planning. Macharis and Bontekoning [6] intercrosses the decision horizon and the decision maker horizon (drayage, terminal, network, and intermodal operators) to classify existing n
Corresponding author.
http://dx.doi.org/10.1016/j.cor.2015.09.009 0305-0548/& 2015 Elsevier Ltd. All rights reserved.
works and identify potential applications of OR methods to intermodal transportation. In addition to adopting the decision horizon, Bektas and Crainic [7] stress the importance of intermodal terminals that form the most critical components of the entire intermodal transportation chain. The efficiency of the latter heavily depends on the speed and reliability of the operations performed by the former. Adopting the decision horizon, SteadieSeifi et al. [8] depicted multimodal decision-making problems in each of the categories respectively with a number of features. They found that the terms multimodal and intermodal are used interchangeably in the literature and thus use multimodal consistently in their survey. However, as mentioned above, some still recognize intermodal transportation as a special case of multimodal transportation. The difference lies in whether the goods themselves are handled during transfers between modes. Multimodal problems appear more complicated than intermodal because translation of flow units between different modes is required. SteadieSeifi et al. [8] focused on the decision planning problem in infrastructure investment decisions, the setup of multimodal terminals, a fundamental problem underlying other management issues. Intermodal terminals constitute the foundation of an intermodal transportation network. This network design problem is usually formulated as a hub location problem, which is also the main body of the decision planning problems in [8]. In a many-to-many distribution system, hubs usually aggregate flows from different origins and dispatch them to different destinations through other hubs. The hub location problem concerns itself with locating hubs and routing flows of origin-destination
114
C.-C. Lin, S.-W. Lin / Computers & Operations Research 67 (2016) 113–119
pairs to pass through them. Surveying the literature before 2008, Alumur and Kara [9] identified two basic types of allocating flows to hubs: single allocation and multiple allocation. In single allocation, all of the outgoing/incoming flow of an origin/destination is allocated to a single hub, while in multiple allocation, the outgoing/incoming flow of an origin/destination can be distributed among multiple hubs. Moreover, hub location problem generally assumes that the network between hubs is a complete graph and that links between hubs cost less than others links. Additionally, allowance of direct flows between non-hub nodes depends on the decision-making context. For example, the hub location problem of Santos et al. [10] allows direct flows between non-hub nodes. Furthermore, Alumur and Kara [9] have classified hub location problems into four categories: the p-hub median problem, the hub location problem with fixed costs, the p-hub center problem and the hub covering problem. The p-hub center problem is a minmax type problem, while the hub covering problem is a facility covering problem, in which demand nodes are considered to be covered if they are within a specified distance of a facility. The p-hub median problem minimizes only the total transportation cost and does not consider the costs of the intermodal terminals. The hub location problem with fixed costs can be regarded as an extension of the p-hub median problem. The term p-hub assumes that the number of hubs must be below or, in some cases, exactly equal p. However, depending on the optimal solution, the number of hubs is no longer a constant in the hub location problem with fixed costs. Both the p-hub median problem and the hub location problem with fixed costs can be further divided into single allocation and multiple allocation problems. The hub location problem with fixed costs also concerns itself with the capacities of hubs. Therefore, the p-hub median problem can also be divided into uncapacitated and capacitated problems. Combinations of these features create different problem variations. For example, the hub and spoke networking problem is a single-allocated p-hub median problem with fixed costs. Hub location problems are multiplex. The reader can also refer to a recent survey by Farahani et al. [11]. Unlike the survey of Alumur and Kara [9] that considers only studies on network-type hub location problems, Farahani et al. [11] have reviewed the most recent advances pf all variants of hub location problems from 2007. The intermodal terminal location problem (ITLP) is a special form of the hub location problem. In their pioneering work, Arnold et al. [12] formulated an extension of the p-hub median problem with fixed costs and unlimited capacities. Although the model allows for transportation between non-hub nodes, unimodal and multimodal transportation between each pair of origin and destination are mutually exclusive. Subsequently, Arnold et al. [13] developed another model to improve the inefficient model in [12], by decreasing the large number of 0–1 variables. This p-hub median problem denotes terminals as arcs rather than nodes to reduce the number of decision variables. The uncapacitated hub location problem with fixed costs of Racunica and Wynter [14] allows no direct flow between non-hub nodes. In the p-hub location problem of Limbourg and Jourquin [15], each origin can be assigned to several hubs, and setup cost is considered. Ishfaq and Sox [16] proposed another model which considers more service and cost issues than related models. Uniquely, the problem is a p-hub median problem with fixed costs. However, no capacities are considered. Sörensen et al. [17] recently developed a model for the ITLP and proved it to be NP-hard. They then developed heuristics to solve it. The representative work of Sörensen et al. [17] was classified by SteadieSeifi et al. [8] as a strategic planning problem with multiallocation, direct shipment, and capacitated hubs. However, these features are themselves insufficient to distinguish the ITLP from similar problems.
Lin et al. [18] indicated that single allocation seems to be unrealistic in a transportation network and prohibitive in terms of minimizing transportation costs. Furthermore, terminals are unlikely to have an unlimited capacity. If terminals were to be capacitated, a limited number of hubs and prohibiting of direct flows between non-hub nodes also appear to be unrealistic. They distinguished the ITLP from hub location problems by the following features: 1. 2. 3. 4.
Unconstrained number of terminals; Capacitated terminals usually with setup costs; Multiple allocation between nodes; Direct shipment between non-hub nodes.
The last three characteristics conform to the classification features applied by SteadieSeifi et al. [8] to the work of Sörensen et al. [17]. However, whether the hub number is limited is not a concern of SteadieSeifi et al. [8]. Although the model of Sörensen et al. [17] conforms to all four features of ITLP, Lin et al. [18] have shown that the model can be improved by eliminating some of its redundant variables and constraints. They also developed two matheuristics to obtain solutions very close to the optimal solutions within a short computation time, especially for large problems. Nonetheless, room for improvement remains. The difficulty in solving the 0–1 programming problem lies mainly in the selection of terminals. Given a set of intermodal terminals, routing the flows with respect to the terminals can be solved by a linear programming problem without 0–1 variables. Further, as mentioned earlier, selecting intermodal terminals is crucial to the ITLP, far more important than routing the transportation. The latter task can become weightless after the terminals are determined. With this in mind, this study proposes to separate the selection of terminals from that of transportation routes in a two-stage optimization approach. The rest of this paper is organized as follows: Section 2 reviews existing models for the ITLP. Section 3 presents the proposed twostage approach, along with proofs of its validity. Section 4 then states a modified matheuristic to solve the ITLP more efficiently. Section 5 experimentally compares models and heuristics. Conclusions and recommendations for future research are offered in Section 6.
2. Models for the ITLP Arnold et al. [12] developed a core model with many extensions to determine the optimal rail/road network in Belgium, which represents a typical intermodal transportation decision-making problem. Most of the mixed 0–1 programming models are p-hub median problems. One of the extensions is a hub location problem with fixed costs, which was modified by Sörensen et al. [17] in developing their own extension. Later, Arnold et al. [13] developed another model for the same problem. That study viewed the terminals as arcs rather than nodes, allowing for a significant reduction in the number of decision variables. Following the course of Arnold et al. [13], Ishfaq and Sox [16] developed a sophisticated model to allow for more practical considerations. Sörensen et al.[17] recently developed another model for the ITLP. Their model is as follows. Let I denote the set of all origin (destination) nodes and K denote the set of all potential terminals in the network such that I \ K ¼ ∅. Each pair of origin and destination nodes ði; jÞ is associated with a transportation cost cij and a demand qij that should be transported. Without a loss of generality, assume that qii ¼ 0. The term wij represents the direct represents the goods transtransport from node i to j while xkm ij ported from node i to j through terminals k and m, respectively.
C.-C. Lin, S.-W. Lin / Computers & Operations Research 67 (2016) 113–119
Associated unit transportation costs are cij and ckm ij ¼ cik þ ckm þ cmj . Each terminal k A K is associated with a capacity of operation C k and a fixed setup cost F k . The 0–1 variable yk signals the state of terminal k, open or not. Terminal k is open when yk ¼ 1 and closed when yk ¼ 0. The definition of xkm ij implies that Sörensen et al. [17] assume that transportation cost is proportional to distance and that the unit transportation cost between the nodes of K is lower than that outside K. Therefore, goods can only be transported directly from i to j or through ðk; mÞ. Other transportation pattern cannot happen because of triangle inequality. The model is given below. Model Sörensen X X X X km ckm cij wij þ F k yk Min ij xij þ i;j A I k;m A K
i;j A I
subject to xkm ij rqij yk ; xkm ij rqij ym ; wij þ
X
xkm ij ¼ qij ;
3. Proposed approach
8 k; m A K;
8 i; jA I;
ð2Þ
8 k; m A K;
ð3Þ
8 i; j A I;
ð4Þ
k;m A K
XX
xkm ij þ
i;j A I m A K
XX
xmk ij r C k ;
8 k A K;
ð5Þ
i;j A I m A K
yk A f0; 1g: Constraints (2) and (3) ensure that no goods can be transported through an unopened terminal. Constraint (4) requires that the transportation demand of each origin/destination pair must be satisfied, through means of unimodal and intermodal transportations in total. Constraint (5) stipulates that the amount of goods transported in and out of terminal k together should not exceed its capacity. By defining flows wij and xkm in terms of real numbers ij rather than 0–1 variables, the Sörensen model appears to be a more practical approach to the ITLP. The formulation of Sörensen et al. [17] conforms to all of the characteristics of ITLP. However, the Sörensen model is still difficult to solve, which explains why that study attempted to solve the model with heuristics. A series of test problems for testing their heuristics were also designed. The Sörensen model, however, can only solve half of the test problems, making it impossible to determine the effectiveness of the heuristics. The difficulty seems to result from the large number of constraints in the Sörensen model, especially those stemming from Constraints (2) and (3). Lin et al. [18] have proved that Constraints (2), (3) and (5) can be replaced by exactly one constraint, namely Constraint (6) in the following model (Model LCL for short). This reduction makes the revised model more efficient than its predecessors. Model LCL X X X X km Min ckm cij wij þ F k yk ij xij þ i;j A I k;m A K
subject to wij þ
i;j A I
X
ð10 Þ
kAK
xkm ij ¼ qij ;
ð40 Þ
8 i; jA I;
k;m A K
XX
xmk ij þ
i;j A I m A K
XX
tolerable time and provide solutions very close to the optimal solutions. Although it is the most efficient model to date, room for improvement remains. Heuristics often evolve with corresponding decision models. Therefore, if using heuristics is inevitable for solving difficult 0–1 programming problems, one approach to improving a heuristic is through fundamental improvement of its decision model. As mentioned earlier, selecting intermodal terminals plays a fundamental role in the model, more important than determining the routes through which goods are transported. If intermodal terminals can be determined before specific routes can be selected accordingly to transport goods, it is very likely that the model for the ITLP can be further improved.
ð1Þ
kAK
8 i; j A I;
115
xkm ij r C k yk ;
8 k A K;
ð6Þ
i;j A I m A K
yk A f0; 1g: However, the model remains a difficult mixed 0–1 programming problem. Lin et al. [18] therefore proposed corresponding matheuristics to solve the problem. The matheuristics can solve problems larger than 200 customers by 200 terminals in a
To reduce the complexity of the state-of-the-art model, assume that the goods shipped from each customer are unique, unlike those of customers. The ITLP can be viewed as a transshipment problem involving multiple goods, in which the customers are sources as well as destinations, and the terminals are transshipment nodes. Let zuv i denote the goods provided by customer i and transported through route ðu; vÞ, u; v A U ¼ I þ K. The proposed multiple transshipment model for the ITLP may then be formulated as follows. Model Proposed X XX cuv zuv F k yk Min i þ i A I u;v A U
subject to
X uAU
X
zuj i
X vAU
X
ziv i
vAU
X
ð7Þ
kAK
zjv i ¼ qij ;
zui i ¼
uAU
zkv i
vAU
XX
X
8 i; jA I; j ai;
X qij ;
ð8Þ
8 i A I;
ð9Þ
8 i A I; k A K;
ð10Þ
jAI
zuk i ¼ 0;
uAU
zmk i þ
iAI mAK
XX
zkm i r C k yk ;
8 k A K;
ð11Þ
iAI mAK
yk A f0; 1g: Objective function (7) calculates the total cost as Function (10 ) does. The demand constraint (8) states that the net flow of goods i into each customer j must equal its demand for those goods. Without loss of generality, assume that qii ¼ 0; 8 i A I. The supply constraint (9) states that the net flow out of customer i (namely, the supply of goods i) must fulfill the total demand for goods i by other customers. Constraint (10) is the balance constraint for the terminal nodes, which requires that no goods can be generated by or accumulated at terminal nodes. Similar to Eq. (6), Eq. (11) requires that the total flow into and out of terminal k be confined by the operating capacity. With the new definition and formulation, the following lemmas can be derived. Lemma 1. When ja i, zjv i ¼ 0; 8 j A I; v A U. Proof. : Case 1. : Constraint (9) has required that customer i fulfill the total demand for goods i by other customers. Excess provision of goods i by customers other than i would incur extra costs. In other words,
116
C.-C. Lin, S.-W. Lin / Computers & Operations Research 67 (2016) 113–119
a zjv i that involves flow originating at node j is unnecessary or forbidden. Case 2. : As Case 1 has addressed the case in which no node can provide goods i, the source can only be node i. When zjv i involves flow from node i, the transportation cost from i to v would be cij þ cjv , which is larger than civ due to triangle inequality. Therefore, transporting goods i through node j to v is unnecessary. The discussions in Cases 1 and 2 lead to the only result in which zjv i ¼ 0 if ja i. QED. Lemma 2. zui i ¼ 0; 8 u A U. Proof. : by Lemma 1, customer i is the only node providing ui goods i. If zui i a 0, there must be a flow equal to zi directly or indirectly transported from node i and back to it. This cycle flow contributes nothing to demand fulfillment and merely increases transportation cost. Therefore, no goods i flow back to node i. QED. An immediate inference of Lemmas 1 and 2 is that Constraints (8) and (9) can be simplified to become (12) and (13). X uj zi ¼ qij ; 8 i; j A I; ð12Þ uAU
X
X qij ;
ziv i ¼
vAU
8 iA I:
ð13Þ
jAI
Other lemmas helpful for proving the equivalence between the proposed model and Model LCL exist.
Lemma 6.
j A I k;m A K
Proof. As Lemma 1 stimulates that no node except i can provide goods i, ziji involves only goods moving directly from node i to j. QED. Lemma 4.
P P j A I k;m A K
xkm ij ¼
P kAK
zik i ; 8 iA I.
Proof. The total goods provided by customer i, according to Constraint (40 ), is X X X X qij ¼ wij þ xkm ij ; 8 i A I: jAI
jAI
j A I k;m A K
On the other hand, since U ¼ I þ K, Eq. (13) can be formulated as, X ij X X qij ¼ zi þ zik i ; 8 i A I: jAI
jAI
kAK
jAI
mAK
P
Lemma 5.
P P j A I k;m A K
xkm ij ¼
P P j A Im A K
P k;m A K
zkm i ; 8 i A I.
mAK
jAI
zjk i
¼ zik i
by Lemma 1, there is Since jAI X kj X X zi þ zkm ¼ zik zmk i i þ i ; 8 i A I; k A K: jAI
mAK
mAK
m0k are mutually exclusive. If cik o cim0 þcm0 k , The terms zik i and zi flow from i through m0 to k cannot happen. Then goods would be transported directly from node i to k. Should this be true, zkj i would be zero because cik þ ckj 4 cij , and transportation would be through route ðk; mÞ or directly from i to j. Thus, the equation reduces to X zkm ¼ zik i i ; 8 i A I; k A K; mAK
and by Lemma 4, X X X X zkm ¼ zik xkm i i ¼ ij ; k;m A K
kAK
j A I k;m A K
Otherwise, if goods should be transported through route ðm km ; kÞ instead of ðk; mÞ, one has zik i ¼ 0 and zi . The equation becomes X kj X zi ¼ zmk i ; 8 iA I; k A K; jAI
mAK
and thus by Lemma 5, X X kj X X X xkm zi ¼ zmk ij ¼ i ; kAK jAI
m;k A K
QED. The equivalence between the proposed model and Model LCL can be verified by the following propositions. Proposition 1. Objective functions (7) and (10 ) measure the same cost. Proof. : Since fðu; vÞj u; v A Ug ¼ fði; jÞj i; jA Igþfðk; mÞj k; m A Kgþfði; kÞj iA I; k A Kg þfðm; jÞj m A K; j A Ig. The total cost to transport goods i is X X X X ij cik zik ckm zkm cmj zmj cij zi ; 8 i A I: i þ i þ i þ kAK
k;m A K
m A K;j A I
jAI
By Lemmas 3–6, it can be replaced by X X X X X X X cik xkm ckm xkm cmj xkm cij wij ; 8 i A I: ij þ ij þ ij þ j A I k;m A K
By Lemma 3, Lemma 4 is proved. QED.
xkm ij ¼
Proof. By replacing U with I þK, Constraint (10) becomes X kj X X jk X zi þ zkm ¼ zi þ zmk i i ; 8 i A I; k A K:
j A I k;m A K
Lemma 3. wij ¼ ziji .
P P
j A I k;m A K
j A I k;m A K
zmj i ; 8 i A I.
Proof. According to Constraint (40 ), X X X X qij ¼ wij þ xkm ij ; 8 i A I: jAI
jAI
j A I k;m A K
Meanwhile, Eq. (12) leads to XX sj X X mj X qij ¼ zi þ zi ; 8 i A I: jAI sAI
jAI
jAI mAK
By Lemma 1, zsj i ¼ 0 when s a i, hence X X mj X X km xij ¼ zi ; 8 iA I: j A I k;m A K
QED.
P sAI
ij zsj i ¼ zi ¼ wij . one has
jAI mAK
Fig. 1. The algorithm of Matheuristic-1.
jAI
C.-C. Lin, S.-W. Lin / Computers & Operations Research 67 (2016) 113–119
117
By the definition that ckm ij ¼ cik þ ckm þ cmj , the total cost to transport all the goods, namely, Objective function (7), equals X X XX X km ckm cij wij þ F k yk ; ij xij þ i A I j A I k;m A K
i;j A I
kAK
which is exactly the same as Objective function (10 ). QED. Proposition 2. Constraints (11) equals Constraint (6). P mk P km xij ¼ zkm and hence xij ¼ zmk Proof. Apparently, i i . Then jAI
XX
zmk i þ
XX
iAI mAK
equals XX
jAI
zkm i r C k yk ;
iAI mAK
xmk ij þ
i;j A I m A K
XX
xkm ij r C k yk :
i;j A I m A K
QED.
Fig. 2. The proposed matheuristic. Table 1 Computation results by different models. Problem
Optimal objective value (107 )
Model LCL Objective RPD
110C_110L_1 110C_110L_2 110C_110L_3 110C_110L_4 110C_110L_5 120C_120L_1 120C_120L_2 120C_120L_3 120C_120L_4 120C_120L_5 130C_130L_1 130C_130L_2 130C_130L_3 130C_130L_4 130C_130L_5 140C_140L_1 140C_140L_2 140C_140L_3 140C_140L_4 140C_140L_5 150C_150L_1 150C_150L_2 150C_150L_3 150C_150L_4 150C_150L_5 160C_160L_1 160C_160L_2 160C_160L_3 160C_160L_4 160C_160L_5 170C_170L_1 170C_170L_2 170C_170L_3 170C_170L_4 170C_170L_5 180C_180L_1 180C_180L_2 180C_180L_3 180C_180L_4 180C_180L_5 190C_190L_1 190C_190L_2 190C_190L_3 190C_190L_4 190C_190L_5 200C_200L_1 200C_200L_2 200C_200L_3 200C_200L_4 200C_200L_5
1495.37 1538.21 1410.20 1496.29 1504.45 1744.64 1837.11 1745.44 1787.36 1731.63 2214.97 2080.85 2111.28 2101.57 2025.77 2411.76 2441.94 2453.64 2399.02 2428.05 2880.42 2868.06 2804.98 2734.99 2739.35 3131.45 3307.49 3243.72 3160.18 3149.45 3667.40 3668.45 3761.57 3674.25 3641.42 3880.67 4088.36 3955.60 4035.32 4074.69 4495.56 4388.51 4460.18 4431.07 4665.39 4971.06 4935.51 4988.04 5059.23 4963.08
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0007% 0.0029% 0 0 0 0.0003% 0 0.1388% 0 0 0 0 0.0017% 0 0 0 0 0 0 0 0 0 0 0 0 0
Proposed model Time(s) 127.68 12122.99 1658.94 11871.55 4934.80 2961.01 639.22 3092.61 26821.12 9778.38 32778.93 31860.87 39758.12 7939.11 269.34 370.21 38758.95 34077.84 17439.27 31339.56 22678.05 443,200 4 43,200 9520.43 4 43,200 4 43,200 15923.96 41862.29 40925.53 4 43,200 4 43,200 4 43,200 26615.90 4 43,200 28663.62 4 43,200 4 43,200 4 43,200 4 43,200 31272.24 2436.15 4 43,200 37026.93 2765.22 4 43,200 3367.65 4 43,200 4 43,200 4 43,200 4 43,200
Objective RPD
Time I (s)
Time II (s)
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
210.48 233.76 391.54 281.88 213.67 317.61 282.28 314.46 321.66 416.07 2011.95 502.57 441.59 457.27 447.42 534.20 686.70 892.64 665.12 667.84 896.97 933.08 832.61 805.40 908.65 3166.34 1154.51 1743.27 1124.00 1130.34 1614.98 1373.49 1218.71 1512.30 2125.96 2182.87 2211.79 2255.13 1893.83 1944.22 2898.67 2394.14 2611.70 2565.92 3402.05 3032.81 3637.06 2993.43 3223.69 3388.23
41.75 46.61 39.30 35.97 45.82 53.09 62.96 61.26 50.87 72.33 176.32 80.17 93.07 74.79 94.99 176.04 219.93 204.22 234.34 108.33 273.58 334.01 290.24 292.55 210.27 288.68 351.98 261.35 275.24 272.40 533.55 407.80 423.63 470.40 522.91 452.75 545.88 758.31 501.91 624.89 1052.77 637.40 618.03 622.54 890.13 848.68 739.49 677.92 1158.42 932.53
Total Time 252.23 280.38 430.83 317.85 259.49 370.69 345.23 375.72 372.53 488.40 2188.26 582.73 534.66 532.06 542.41 710.24 906.63 1096.86 899.46 776.17 1170.54 1267.09 1122.85 1097.95 1118.93 3455.02 1506.49 2004.62 1399.24 1402.74 2148.53 1781.29 1642.33 1982.70 2648.87 2635.63 2757.67 3013.45 2395.73 2569.11 3951.44 3031.53 3229.73 3188.45 4292.18 3881.49 4376.55 3671.35 4382.11 4320.76
118
C.-C. Lin, S.-W. Lin / Computers & Operations Research 67 (2016) 113–119
The equality between Model LCL and the proposed model has been proved. The proposed model may now be used to determine the optimal terminal set, which should be identical to that found using Model LCL, except that exactly how the goods are transported from i to j, namely, the value of xkm ij , is unavailable. xkm ij
can be obtained using Model LCL after the Nevertheless, terminals have been determined, if the decision-maker needs to know them. This constitutes a two-stage optimization approach for the ITLP. The difficulty of solving the ITLP now lies in the first stage, which is a mixed 0–1 programming problem. The terminals being determined, the second stage is simply a typical transshipment problem without 0–1 variables. With fewer decision variables, the proposed model is expected to be more efficient than Model LCL. The number of decision variables in Model LCL is j Ij 2 j K j 2 þ j Ij 2 þ j K j while that of the proposed model is only j Ij j I þ K j 2 þ j K j . For example, when
j Ij ¼ j K j ¼ 100, Model LCL would use 25 times more variables than those used by the proposed model.
4. Heuristic The above propositions have shown that with fewer decision variables, the proposed model can find the optimal terminals as those found by using the model of Lin et al. [18]. In addition to proposing a revised model for the ITLP, they also proposed two matheuristics, which have been shown to be capable of giving near optimal solutions with less computation time. Among the two matheuristics, the first matheuristic (Matheuristic-1) performs better. Fig. 1 shows the algorithm of Matheuristic-1. After solving the relaxed Model LCL, Matheuristic-1 sets the yk of all the partially utilized terminals to be 0–1 variables and solves the model again. A drawback of this matheuristic is that the number of
Table 2 Computation results by different matheuristics. Problem
110C_110L_1 110C_110L_2 110C_110L_3 110C_110L_4 110C_110L_5 120C_120L_1 120C_120L_2 120C_120L_3 120C_120L_4 120C_120L_5 130C_130L_1 130C_130L_2 130C_130L_3 130C_130L_4 130C_130L_5 140C_140L_1 140C_140L_2 140C_140L_3 140C_140L_4 140C_140L_5 150C_150L_1 150C_150L_2 150C_150L_3 150C_150L_4 150C_150L_5 160C_160L_1 160C_160L_2 160C_160L_3 160C_160L_4 160C_160L_5 170C_170L_1 170C_170L_2 170C_170L_3 170C_170L_4 170C_170L_5 180C_180L_1 180C_180L_2 180C_180L_3 180C_180L_4 180C_180L_5 190C_190L_1 190C_190L_2 190C_190L_3 190C_190L_4 190C_190L_5 200C_200L_1 200C_200L_2 200C_200L_3 200C_200L_4 200C_200L_5
Optimal objective value (107 )
1495.37 1538.21 1410.20 1496.29 1504.45 1744.64 1837.11 1745.44 1787.36 1731.63 2214.97 2080.85 2111.28 2101.57 2025.77 2411.76 2441.94 2453.64 2399.02 2428.05 2880.42 2868.06 2804.98 2734.99 2739.35 3131.45 3307.49 3243.72 3160.18 3149.45 3667.40 3668.45 3761.57 3674.25 3641.42 3880.67 4088.36 3955.60 4035.32 4074.69 4495.56 4388.51 4460.18 4431.07 4665.39 4971.06 4935.51 4988.04 5059.23 4963.08
Matheuristic-1
Proposed matheuristic
Objective RPD
Time I (s)
Time II (s)
Total Time
Objective RPD
Time I (s)
Time II (s)
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0003% 0 0 0 0 0 0 0 0 0.0003% 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
83.21 275.43 573.00 248.34 188.05 521.92 319.90 365.52 362.11 789.78 1121.15 589.14 480.03 1060.94 176.18 266.42 667.13 789.27 1065.85 706.82 846.49 972.87 919.96 900.39 1076.57 1931.74 1255.20 1867.51 1397.43 7286.20 2962.16 1473.18 1458.86 1507.79 1620.72 1860.17 2183.88 2022.31 1699.72 1704.81 801.92 2499.79 4409.09 2176.28 2234.21 854.59 2750.72 2545.20 2985.47 3096.03
50.47 47.11 44.22 37.26 47.55 46.79 74.02 63.58 51.80 59.90 170.24 77.52 95.72 86.37 99.37 193.52 226.80 187.89 231.21 110.37 279.20 321.40 290.30 298.59 216.40 307.47 351.03 311.04 285.52 282.76 532.98 408.17 434.28 499.60 522.33 473.37 542.83 596.12 423.72 685.78 784.49 807.76 623.26 632.26 693.40 1108.88 753.70 689.38 1242.18 1166.1650
133.68 322.54 617.22 285.60 235.60 568.71 393.92 429.10 413.90 849.67 1291.40 666.66 575.75 1147.31 275.55 459.94 893.93 977.16 1297.06 817.19 1125.69 1294.27 1210.25 1198.98 1292.97 2239.21 1606.23 2178.56 1682.95 7568.96 3495.14 1881.35 1893.14 2007.39 2143.05 2333.54 2726.71 2618.43 2123.44 2390.59 1586.41 3307.55 5032.35 2808.54 2927.60 1963.47 3504.42 3234.59 4227.65 4262.19
0 0 0 0 0.0007% 0.0017% 0 0 0 0 0 0.0005% 0.0014% 0 0 0 0 0.0004% 0 0 0 0 0 0 0 0 0.0003% 0 0 0 0 0 0 0 0 0.0003% 0 0.0003% 0 0 0 0 0 0.0002% 0 0 0 0.0004% 0.0006% 0.0002%
117.64 59.18 87.77 100.17 102.45 69.62 70.41 139.06 88.69 77.61 240.58 215.81 197.47 210.09 230.56 314.66 324.36 343.42 311.83 288.39 398.55 343.82 382.47 431.01 423.58 500.82 602.83 499.62 549.17 518.27 310.70 644.10 719.70 650.02 617.31 870.71 761.08 832.34 839.98 793.73 847.70 997.30 897.19 837.65 886.83 1418.04 1073.61 1149.78 1270.39 1182.37
47.43 46.41 37.39 36.88 46.86 54.44 70.18 68.35 48.08 67.48 152.62 88.01 107.26 82.34 97.02 179.98 220.08 172.44 243.73 109.08 275.04 331.98 281.26 304.55 212.74 332.44 356.23 301.37 298.18 273.68 529.90 522.20 480.24 494.38 465.11 586.11 544.36 582.40 404.45 567.96 792.26 938.21 628.93 589.35 712.35 1157.62 750.70 848.25 1094.56 1066.16
Total Time 165.07 105.58 125.16 137.06 149.30 124.07 140.60 207.40 136.76 145.09 393.20 303.82 304.73 292.43 327.58 494.64 544.44 515.86 555.55 397.47 673.59 675.81 663.73 735.56 636.32 833.26 959.06 800.99 847.35 791.95 840.59 1166.30 1199.94 1144.40 1082.41 1456.81 1305.44 1414.74 1244.43 1361.69 1639.96 1935.51 1526.12 1427.00 1599.18 2575.67 1824.31 1998.03 2364.94 2248.53
C.-C. Lin, S.-W. Lin / Computers & Operations Research 67 (2016) 113–119
required iterations that solve a series of mixed 0–1 programming models to reach the final solution is unpredictable. A long computation time might be required as the problem is complex. One possible approach to controlling the computation time of the unpredictable Matheuristic-1 is to set all of the remaining yk with a value less than 1 to be 0–1 variables after the initial relaxed model is solved. In this way, iterative solving of 0–1 programming models is avoided. Meanwhile, all direct flows between customers and those through terminals with yk ¼ 1 are fixed when solving the following 0–1 programming model. Fig. 2 states the algorithm of the proposed matheuristic. The two matheuristics will be compared in the following section.
5. Computational results This section compares the proposed model with that of Lin et al. [18], as well as the proposed matheuristic with their Matheuristic-1. The test problems comprise only the larger problems in [18] because the smaller problems that range from 10–100 customers/terminals appeared to be too easy to effectively distinguish between the models. The details of generating the test problems can be found in [18]. The test problems were solved using Gurobi 5.5 on a workstation with two 2.6 GHz Intel Xeon E52670 CPUs and 96 GB RAM, which is the same as in [18]. The allowable computation time was also set to 12 h. Table 1 summarizes the computational results by the models. The solution quality is measured using relative percentage deviation (RPD) with respect to the optimal objective value. Time I and II represent respectively the times of Stage I and II. On average, the proposed model used about only 5% of the computation time used by Model LCL, and all of the test problems were solved within the time limit. Although Model LCL has found the optimal solutions to some of the large problems before the time was out, the optimality for these problems was undetermined. The results demonstrate that the proposed model with largely reduced decision variables is significantly more efficient than existing models for the ITLP. Less computation time is required when selecting intermodal terminals, as mentioned above, is the only concern of the decisionmaker. If this is the case, the computation time can be further reduced by 20% because Time II is roughly 25% of Time I. Table 2 compares the proposed matheuristic with Matheuristic-1, both based on the proposed model. The proposed matheuristic uses approximately 50% of the computation time used by Matheuristic-1. A paired t-test suggests that the computation time difference is statistically significant. Though the proposed matheuristic appears more efficient than Matheuristic-1, the improvement is not as significant as that achieved by the proposed model, indicating that the use of matheuristics seems to have reached a limit. Nevertheless, the proposed matheuristic is still helpful. Comparing the results in Tables 1 and 2 reveals that Matheuristic-1 is very unlikely to be helpful in reducing computation time. The computation time of Matheuristic-1 is even slightly longer than that of the proposed model. This suggests that Matheuristic-1 is obsolete after the new model is proposed. As mentioned earlier, this obsolescence may result from the repeated solving of a long series of 0–1 programming problems. Although the proposed matheuristic is slightly less sophisticated than Matheuristic-1 when it comes to optimality, the objective RPD remains at a very low level.
6. Conclusions Multimodal transportation has recently attracted much attention. Among related problems, the intermodal terminal location problem (ITLP) aims to determine the locations of intermodal terminals.
119
According to Lin et al. [18], the ITLP represents a more realistic formulation than others based on hub location problems. Sörensen et al. [17] identified this problem and provided a mixed 0–1 programming model for it. Though Lin et al. [18] have revised the model to make it more efficient, the potential for further refinement remains. This study proposes selecting intermodal terminals first and then routing the flows between customers. In the first stage, exact flows between customer pairs are aggregated into path flows, and decision variables can be largely reduced, accordingly. After the terminal nodes are selected, in the second stage there remains only a simple linear programming problem. The second stage may be omitted if the decision-maker's only concern is the terminal locations. Experimental results show that the improvement via this approach is significant. The computation time can be shortened by around 95%. However, when attempting to further reduce the computation time by using the Matheuristic-1 of Lin et al. [18], no significant improvement can be observed. Therefore, a new matheuristic was devised, in order to reduce the computation time while maintaining the solution quality. The new matheuristic did reduce the computation time by half, but the improvement was not very significant. This fact indicates that the refinement of programming model, and thus the capability of matheuristics, has reached a limit. To more rapidly obtain the solution, other heuristics or metaheuristics might offer better odds.
Acknowledgment The authors thank the Ministry of Science and Technology, Republic of China (Taiwan) for financially supporting this study under Grant no. MOST103-2410-H-182-006.
References [1] European Conference of Ministers of Transport. Terminology on combined transport. 〈http:/www.cordis.lu/transport〉; 1993. [2] Chang T-S. Best routes selection in international intermodal networks. Comput Oper Res 2008;35:2877–91. [3] Hanssen T-ES, Mathisen TA, Jørgensen F. Generalized transport costs in intermodal freight transport. Proced Social Behav Sci 2012;54:189–200. [4] Janic M. Modelling the full costs of an intermodal and road freight transport network. Transp Res Part D 2007;12:33–44. [5] Caris A, Macharis C, Janssens GK. Decision support in intermodal transport: a new research agenda. Comput Ind 2013;64:105–12. [6] Macharis C, Bontekoning YM. Opportunities for OR in intermodal freight transport research: a review. Eur J Oper Res 2004;153:400–16. [7] Bektas T, Crainic T. A brief review of intermodal transportation. In: Taylor G, editor. Logistics engineering handbook. London: CRC Press; 2007. p. 1–16. [8] SteadieSeifi M, Dellaert NP, Nuijten T, Van Woensel T, Raoufi R. Multimodal fright transportation planning: a literature review. Eur J Oper Res 2014;233:1–15. [9] Alumur S, Kara BY. Network hub location problems: the state of the art. Eur J Oper Res 2008;190:1–21. [10] Santos BF, Limbourg S, Carreira JS. The impact of transport policies on railroad intermodal freight competitiveness – the case of Belgium. Transp Res Part D 2015;34:230–44. [11] Farahani RZ, Hekmatfar M, Arabani AB, Nikbakhsh E. Hub location problems: a review of models, classification, solution techniques, and applications. Comput Ind Eng 2013;64:1096–109. [12] Arnold P, Peeters D, Thomas I, Marchand H. Pour une localisation optimale des centre de transbordement intermodaux entre réseaux de transport: formulation et extensions. Can Geogr 2001;45:427–36. [13] Arnold P, Peeters D, Thomas I. Modelling a rail/road intermodal transportation system. Transp Res Part E 2004;40:255–70. [14] Racunica I, Wynter L. Optimal location of intermodal freight hubs. Transp Res Part B 2005;39:453–77. [15] Limbourg S, Jourquin B. Optimal rail-road container terminal locations on the European network. Transp Res Part E 2009;45:551–63. [16] Ishfaq R, Sox CR. Intermodal logistics: the interplay of financial, operational and service issues. Transp Res Part E 2010;46:926–49. [17] Sörensen K, Vanovermeire C, Busschaert S. Efficient metaheuristics to solve the intermodal terminal location problem. Comput Oper Res 2012;39:2079–90. [18] Lin CC, Chiang YI, Lin SW. Efficient model and heuristics for the intermodal terminal location problem. Comput Oper Res 2014;51:41–51.
Contents lists available at ScienceDirect
Computers & Operations Research journal homepage: www.elsevier.com/locate/caor
Two-stage approach to the intermodal terminal location problem Chang-Chun Lin, Shih-Wei Lin n Department of Information Management, Chang Gung University, 259 Wen-Hwa 1st Road, Kwei-Shan, Tao-Yuan 333, Taiwan, ROC
art ic l e i nf o
Keywords: Multimodal transportation Intermodal transportation Intermodal terminals 0–1 Programming Matheuristics
a b s t r a c t Multimodal transportation means to transport freight using at least two transportation modes. Intermodal freight transportation, a particular form of multimodal transportation, transports freight in an intermodal container without handling of the freight itself when changing modes. The intermodal terminal location problem involves selecting terminals that constitute an intermodal transportation network and routing freight flows with minimal total transportation and operating costs. Arnold et al. first presented mathematical programming models for the problem. Sörensen et al. recently proposed another model for the problem. However, Lin et al. have shown that the model of Sörensen et al. is complex and time-consuming and therefore developed a modified mixed integer programming model to increase computation efficiency. This study shows that the model of Lin et al. can be further improved by separating the selection of intermodal terminals from the routing of transportation flows. A two-stage programming approach is proposed along with a modified, more efficient matheuristic. & 2015 Elsevier Ltd. All rights reserved.
1. Introduction As global trade expands, long-haul transportation is increasingly important. Unimodal long-haul transportation that uses only one transportation mode (e.g., road, rail, or water) is literally impossible for global trade. Managing multimodal transportation is thus important both practically and academically. Multimodal transportation transports goods by using at least two transportation modes. Intermodal transportation, a particular form multimodal transportation, has received considerable attention in recent years. According to the European Conference of Ministers of Transport [1], intermodal transportation refers to the movement of goods in the same loading unit without handling the goods themselves during transfers between modes. Intermodal transportation is well recognized as environmentally friendly, capable of reducing congestion, accessible, and highly feasible for global trading. The multi-faceted literature on multimodal and intermodal transportation addresses issues such as decision of the number and location of terminals, route selection [2], cost analysis [2,3], transport policy [4] and intermodal decision support [5]. Literature reviews on multimodal transportation followed a common framework of the decision horizon of planning: strategic, tactical, and operational planning. Macharis and Bontekoning [6] intercrosses the decision horizon and the decision maker horizon (drayage, terminal, network, and intermodal operators) to classify existing n
Corresponding author.
http://dx.doi.org/10.1016/j.cor.2015.09.009 0305-0548/& 2015 Elsevier Ltd. All rights reserved.
works and identify potential applications of OR methods to intermodal transportation. In addition to adopting the decision horizon, Bektas and Crainic [7] stress the importance of intermodal terminals that form the most critical components of the entire intermodal transportation chain. The efficiency of the latter heavily depends on the speed and reliability of the operations performed by the former. Adopting the decision horizon, SteadieSeifi et al. [8] depicted multimodal decision-making problems in each of the categories respectively with a number of features. They found that the terms multimodal and intermodal are used interchangeably in the literature and thus use multimodal consistently in their survey. However, as mentioned above, some still recognize intermodal transportation as a special case of multimodal transportation. The difference lies in whether the goods themselves are handled during transfers between modes. Multimodal problems appear more complicated than intermodal because translation of flow units between different modes is required. SteadieSeifi et al. [8] focused on the decision planning problem in infrastructure investment decisions, the setup of multimodal terminals, a fundamental problem underlying other management issues. Intermodal terminals constitute the foundation of an intermodal transportation network. This network design problem is usually formulated as a hub location problem, which is also the main body of the decision planning problems in [8]. In a many-to-many distribution system, hubs usually aggregate flows from different origins and dispatch them to different destinations through other hubs. The hub location problem concerns itself with locating hubs and routing flows of origin-destination
114
C.-C. Lin, S.-W. Lin / Computers & Operations Research 67 (2016) 113–119
pairs to pass through them. Surveying the literature before 2008, Alumur and Kara [9] identified two basic types of allocating flows to hubs: single allocation and multiple allocation. In single allocation, all of the outgoing/incoming flow of an origin/destination is allocated to a single hub, while in multiple allocation, the outgoing/incoming flow of an origin/destination can be distributed among multiple hubs. Moreover, hub location problem generally assumes that the network between hubs is a complete graph and that links between hubs cost less than others links. Additionally, allowance of direct flows between non-hub nodes depends on the decision-making context. For example, the hub location problem of Santos et al. [10] allows direct flows between non-hub nodes. Furthermore, Alumur and Kara [9] have classified hub location problems into four categories: the p-hub median problem, the hub location problem with fixed costs, the p-hub center problem and the hub covering problem. The p-hub center problem is a minmax type problem, while the hub covering problem is a facility covering problem, in which demand nodes are considered to be covered if they are within a specified distance of a facility. The p-hub median problem minimizes only the total transportation cost and does not consider the costs of the intermodal terminals. The hub location problem with fixed costs can be regarded as an extension of the p-hub median problem. The term p-hub assumes that the number of hubs must be below or, in some cases, exactly equal p. However, depending on the optimal solution, the number of hubs is no longer a constant in the hub location problem with fixed costs. Both the p-hub median problem and the hub location problem with fixed costs can be further divided into single allocation and multiple allocation problems. The hub location problem with fixed costs also concerns itself with the capacities of hubs. Therefore, the p-hub median problem can also be divided into uncapacitated and capacitated problems. Combinations of these features create different problem variations. For example, the hub and spoke networking problem is a single-allocated p-hub median problem with fixed costs. Hub location problems are multiplex. The reader can also refer to a recent survey by Farahani et al. [11]. Unlike the survey of Alumur and Kara [9] that considers only studies on network-type hub location problems, Farahani et al. [11] have reviewed the most recent advances pf all variants of hub location problems from 2007. The intermodal terminal location problem (ITLP) is a special form of the hub location problem. In their pioneering work, Arnold et al. [12] formulated an extension of the p-hub median problem with fixed costs and unlimited capacities. Although the model allows for transportation between non-hub nodes, unimodal and multimodal transportation between each pair of origin and destination are mutually exclusive. Subsequently, Arnold et al. [13] developed another model to improve the inefficient model in [12], by decreasing the large number of 0–1 variables. This p-hub median problem denotes terminals as arcs rather than nodes to reduce the number of decision variables. The uncapacitated hub location problem with fixed costs of Racunica and Wynter [14] allows no direct flow between non-hub nodes. In the p-hub location problem of Limbourg and Jourquin [15], each origin can be assigned to several hubs, and setup cost is considered. Ishfaq and Sox [16] proposed another model which considers more service and cost issues than related models. Uniquely, the problem is a p-hub median problem with fixed costs. However, no capacities are considered. Sörensen et al. [17] recently developed a model for the ITLP and proved it to be NP-hard. They then developed heuristics to solve it. The representative work of Sörensen et al. [17] was classified by SteadieSeifi et al. [8] as a strategic planning problem with multiallocation, direct shipment, and capacitated hubs. However, these features are themselves insufficient to distinguish the ITLP from similar problems.
Lin et al. [18] indicated that single allocation seems to be unrealistic in a transportation network and prohibitive in terms of minimizing transportation costs. Furthermore, terminals are unlikely to have an unlimited capacity. If terminals were to be capacitated, a limited number of hubs and prohibiting of direct flows between non-hub nodes also appear to be unrealistic. They distinguished the ITLP from hub location problems by the following features: 1. 2. 3. 4.
Unconstrained number of terminals; Capacitated terminals usually with setup costs; Multiple allocation between nodes; Direct shipment between non-hub nodes.
The last three characteristics conform to the classification features applied by SteadieSeifi et al. [8] to the work of Sörensen et al. [17]. However, whether the hub number is limited is not a concern of SteadieSeifi et al. [8]. Although the model of Sörensen et al. [17] conforms to all four features of ITLP, Lin et al. [18] have shown that the model can be improved by eliminating some of its redundant variables and constraints. They also developed two matheuristics to obtain solutions very close to the optimal solutions within a short computation time, especially for large problems. Nonetheless, room for improvement remains. The difficulty in solving the 0–1 programming problem lies mainly in the selection of terminals. Given a set of intermodal terminals, routing the flows with respect to the terminals can be solved by a linear programming problem without 0–1 variables. Further, as mentioned earlier, selecting intermodal terminals is crucial to the ITLP, far more important than routing the transportation. The latter task can become weightless after the terminals are determined. With this in mind, this study proposes to separate the selection of terminals from that of transportation routes in a two-stage optimization approach. The rest of this paper is organized as follows: Section 2 reviews existing models for the ITLP. Section 3 presents the proposed twostage approach, along with proofs of its validity. Section 4 then states a modified matheuristic to solve the ITLP more efficiently. Section 5 experimentally compares models and heuristics. Conclusions and recommendations for future research are offered in Section 6.
2. Models for the ITLP Arnold et al. [12] developed a core model with many extensions to determine the optimal rail/road network in Belgium, which represents a typical intermodal transportation decision-making problem. Most of the mixed 0–1 programming models are p-hub median problems. One of the extensions is a hub location problem with fixed costs, which was modified by Sörensen et al. [17] in developing their own extension. Later, Arnold et al. [13] developed another model for the same problem. That study viewed the terminals as arcs rather than nodes, allowing for a significant reduction in the number of decision variables. Following the course of Arnold et al. [13], Ishfaq and Sox [16] developed a sophisticated model to allow for more practical considerations. Sörensen et al.[17] recently developed another model for the ITLP. Their model is as follows. Let I denote the set of all origin (destination) nodes and K denote the set of all potential terminals in the network such that I \ K ¼ ∅. Each pair of origin and destination nodes ði; jÞ is associated with a transportation cost cij and a demand qij that should be transported. Without a loss of generality, assume that qii ¼ 0. The term wij represents the direct represents the goods transtransport from node i to j while xkm ij ported from node i to j through terminals k and m, respectively.
C.-C. Lin, S.-W. Lin / Computers & Operations Research 67 (2016) 113–119
Associated unit transportation costs are cij and ckm ij ¼ cik þ ckm þ cmj . Each terminal k A K is associated with a capacity of operation C k and a fixed setup cost F k . The 0–1 variable yk signals the state of terminal k, open or not. Terminal k is open when yk ¼ 1 and closed when yk ¼ 0. The definition of xkm ij implies that Sörensen et al. [17] assume that transportation cost is proportional to distance and that the unit transportation cost between the nodes of K is lower than that outside K. Therefore, goods can only be transported directly from i to j or through ðk; mÞ. Other transportation pattern cannot happen because of triangle inequality. The model is given below. Model Sörensen X X X X km ckm cij wij þ F k yk Min ij xij þ i;j A I k;m A K
i;j A I
subject to xkm ij rqij yk ; xkm ij rqij ym ; wij þ
X
xkm ij ¼ qij ;
3. Proposed approach
8 k; m A K;
8 i; jA I;
ð2Þ
8 k; m A K;
ð3Þ
8 i; j A I;
ð4Þ
k;m A K
XX
xkm ij þ
i;j A I m A K
XX
xmk ij r C k ;
8 k A K;
ð5Þ
i;j A I m A K
yk A f0; 1g: Constraints (2) and (3) ensure that no goods can be transported through an unopened terminal. Constraint (4) requires that the transportation demand of each origin/destination pair must be satisfied, through means of unimodal and intermodal transportations in total. Constraint (5) stipulates that the amount of goods transported in and out of terminal k together should not exceed its capacity. By defining flows wij and xkm in terms of real numbers ij rather than 0–1 variables, the Sörensen model appears to be a more practical approach to the ITLP. The formulation of Sörensen et al. [17] conforms to all of the characteristics of ITLP. However, the Sörensen model is still difficult to solve, which explains why that study attempted to solve the model with heuristics. A series of test problems for testing their heuristics were also designed. The Sörensen model, however, can only solve half of the test problems, making it impossible to determine the effectiveness of the heuristics. The difficulty seems to result from the large number of constraints in the Sörensen model, especially those stemming from Constraints (2) and (3). Lin et al. [18] have proved that Constraints (2), (3) and (5) can be replaced by exactly one constraint, namely Constraint (6) in the following model (Model LCL for short). This reduction makes the revised model more efficient than its predecessors. Model LCL X X X X km Min ckm cij wij þ F k yk ij xij þ i;j A I k;m A K
subject to wij þ
i;j A I
X
ð10 Þ
kAK
xkm ij ¼ qij ;
ð40 Þ
8 i; jA I;
k;m A K
XX
xmk ij þ
i;j A I m A K
XX
tolerable time and provide solutions very close to the optimal solutions. Although it is the most efficient model to date, room for improvement remains. Heuristics often evolve with corresponding decision models. Therefore, if using heuristics is inevitable for solving difficult 0–1 programming problems, one approach to improving a heuristic is through fundamental improvement of its decision model. As mentioned earlier, selecting intermodal terminals plays a fundamental role in the model, more important than determining the routes through which goods are transported. If intermodal terminals can be determined before specific routes can be selected accordingly to transport goods, it is very likely that the model for the ITLP can be further improved.
ð1Þ
kAK
8 i; j A I;
115
xkm ij r C k yk ;
8 k A K;
ð6Þ
i;j A I m A K
yk A f0; 1g: However, the model remains a difficult mixed 0–1 programming problem. Lin et al. [18] therefore proposed corresponding matheuristics to solve the problem. The matheuristics can solve problems larger than 200 customers by 200 terminals in a
To reduce the complexity of the state-of-the-art model, assume that the goods shipped from each customer are unique, unlike those of customers. The ITLP can be viewed as a transshipment problem involving multiple goods, in which the customers are sources as well as destinations, and the terminals are transshipment nodes. Let zuv i denote the goods provided by customer i and transported through route ðu; vÞ, u; v A U ¼ I þ K. The proposed multiple transshipment model for the ITLP may then be formulated as follows. Model Proposed X XX cuv zuv F k yk Min i þ i A I u;v A U
subject to
X uAU
X
zuj i
X vAU
X
ziv i
vAU
X
ð7Þ
kAK
zjv i ¼ qij ;
zui i ¼
uAU
zkv i
vAU
XX
X
8 i; jA I; j ai;
X qij ;
ð8Þ
8 i A I;
ð9Þ
8 i A I; k A K;
ð10Þ
jAI
zuk i ¼ 0;
uAU
zmk i þ
iAI mAK
XX
zkm i r C k yk ;
8 k A K;
ð11Þ
iAI mAK
yk A f0; 1g: Objective function (7) calculates the total cost as Function (10 ) does. The demand constraint (8) states that the net flow of goods i into each customer j must equal its demand for those goods. Without loss of generality, assume that qii ¼ 0; 8 i A I. The supply constraint (9) states that the net flow out of customer i (namely, the supply of goods i) must fulfill the total demand for goods i by other customers. Constraint (10) is the balance constraint for the terminal nodes, which requires that no goods can be generated by or accumulated at terminal nodes. Similar to Eq. (6), Eq. (11) requires that the total flow into and out of terminal k be confined by the operating capacity. With the new definition and formulation, the following lemmas can be derived. Lemma 1. When ja i, zjv i ¼ 0; 8 j A I; v A U. Proof. : Case 1. : Constraint (9) has required that customer i fulfill the total demand for goods i by other customers. Excess provision of goods i by customers other than i would incur extra costs. In other words,
116
C.-C. Lin, S.-W. Lin / Computers & Operations Research 67 (2016) 113–119
a zjv i that involves flow originating at node j is unnecessary or forbidden. Case 2. : As Case 1 has addressed the case in which no node can provide goods i, the source can only be node i. When zjv i involves flow from node i, the transportation cost from i to v would be cij þ cjv , which is larger than civ due to triangle inequality. Therefore, transporting goods i through node j to v is unnecessary. The discussions in Cases 1 and 2 lead to the only result in which zjv i ¼ 0 if ja i. QED. Lemma 2. zui i ¼ 0; 8 u A U. Proof. : by Lemma 1, customer i is the only node providing ui goods i. If zui i a 0, there must be a flow equal to zi directly or indirectly transported from node i and back to it. This cycle flow contributes nothing to demand fulfillment and merely increases transportation cost. Therefore, no goods i flow back to node i. QED. An immediate inference of Lemmas 1 and 2 is that Constraints (8) and (9) can be simplified to become (12) and (13). X uj zi ¼ qij ; 8 i; j A I; ð12Þ uAU
X
X qij ;
ziv i ¼
vAU
8 iA I:
ð13Þ
jAI
Other lemmas helpful for proving the equivalence between the proposed model and Model LCL exist.
Lemma 6.
j A I k;m A K
Proof. As Lemma 1 stimulates that no node except i can provide goods i, ziji involves only goods moving directly from node i to j. QED. Lemma 4.
P P j A I k;m A K
xkm ij ¼
P kAK
zik i ; 8 iA I.
Proof. The total goods provided by customer i, according to Constraint (40 ), is X X X X qij ¼ wij þ xkm ij ; 8 i A I: jAI
jAI
j A I k;m A K
On the other hand, since U ¼ I þ K, Eq. (13) can be formulated as, X ij X X qij ¼ zi þ zik i ; 8 i A I: jAI
jAI
kAK
jAI
mAK
P
Lemma 5.
P P j A I k;m A K
xkm ij ¼
P P j A Im A K
P k;m A K
zkm i ; 8 i A I.
mAK
jAI
zjk i
¼ zik i
by Lemma 1, there is Since jAI X kj X X zi þ zkm ¼ zik zmk i i þ i ; 8 i A I; k A K: jAI
mAK
mAK
m0k are mutually exclusive. If cik o cim0 þcm0 k , The terms zik i and zi flow from i through m0 to k cannot happen. Then goods would be transported directly from node i to k. Should this be true, zkj i would be zero because cik þ ckj 4 cij , and transportation would be through route ðk; mÞ or directly from i to j. Thus, the equation reduces to X zkm ¼ zik i i ; 8 i A I; k A K; mAK
and by Lemma 4, X X X X zkm ¼ zik xkm i i ¼ ij ; k;m A K
kAK
j A I k;m A K
Otherwise, if goods should be transported through route ðm km ; kÞ instead of ðk; mÞ, one has zik i ¼ 0 and zi . The equation becomes X kj X zi ¼ zmk i ; 8 iA I; k A K; jAI
mAK
and thus by Lemma 5, X X kj X X X xkm zi ¼ zmk ij ¼ i ; kAK jAI
m;k A K
QED. The equivalence between the proposed model and Model LCL can be verified by the following propositions. Proposition 1. Objective functions (7) and (10 ) measure the same cost. Proof. : Since fðu; vÞj u; v A Ug ¼ fði; jÞj i; jA Igþfðk; mÞj k; m A Kgþfði; kÞj iA I; k A Kg þfðm; jÞj m A K; j A Ig. The total cost to transport goods i is X X X X ij cik zik ckm zkm cmj zmj cij zi ; 8 i A I: i þ i þ i þ kAK
k;m A K
m A K;j A I
jAI
By Lemmas 3–6, it can be replaced by X X X X X X X cik xkm ckm xkm cmj xkm cij wij ; 8 i A I: ij þ ij þ ij þ j A I k;m A K
By Lemma 3, Lemma 4 is proved. QED.
xkm ij ¼
Proof. By replacing U with I þK, Constraint (10) becomes X kj X X jk X zi þ zkm ¼ zi þ zmk i i ; 8 i A I; k A K:
j A I k;m A K
Lemma 3. wij ¼ ziji .
P P
j A I k;m A K
j A I k;m A K
zmj i ; 8 i A I.
Proof. According to Constraint (40 ), X X X X qij ¼ wij þ xkm ij ; 8 i A I: jAI
jAI
j A I k;m A K
Meanwhile, Eq. (12) leads to XX sj X X mj X qij ¼ zi þ zi ; 8 i A I: jAI sAI
jAI
jAI mAK
By Lemma 1, zsj i ¼ 0 when s a i, hence X X mj X X km xij ¼ zi ; 8 iA I: j A I k;m A K
QED.
P sAI
ij zsj i ¼ zi ¼ wij . one has
jAI mAK
Fig. 1. The algorithm of Matheuristic-1.
jAI
C.-C. Lin, S.-W. Lin / Computers & Operations Research 67 (2016) 113–119
117
By the definition that ckm ij ¼ cik þ ckm þ cmj , the total cost to transport all the goods, namely, Objective function (7), equals X X XX X km ckm cij wij þ F k yk ; ij xij þ i A I j A I k;m A K
i;j A I
kAK
which is exactly the same as Objective function (10 ). QED. Proposition 2. Constraints (11) equals Constraint (6). P mk P km xij ¼ zkm and hence xij ¼ zmk Proof. Apparently, i i . Then jAI
XX
zmk i þ
XX
iAI mAK
equals XX
jAI
zkm i r C k yk ;
iAI mAK
xmk ij þ
i;j A I m A K
XX
xkm ij r C k yk :
i;j A I m A K
QED.
Fig. 2. The proposed matheuristic. Table 1 Computation results by different models. Problem
Optimal objective value (107 )
Model LCL Objective RPD
110C_110L_1 110C_110L_2 110C_110L_3 110C_110L_4 110C_110L_5 120C_120L_1 120C_120L_2 120C_120L_3 120C_120L_4 120C_120L_5 130C_130L_1 130C_130L_2 130C_130L_3 130C_130L_4 130C_130L_5 140C_140L_1 140C_140L_2 140C_140L_3 140C_140L_4 140C_140L_5 150C_150L_1 150C_150L_2 150C_150L_3 150C_150L_4 150C_150L_5 160C_160L_1 160C_160L_2 160C_160L_3 160C_160L_4 160C_160L_5 170C_170L_1 170C_170L_2 170C_170L_3 170C_170L_4 170C_170L_5 180C_180L_1 180C_180L_2 180C_180L_3 180C_180L_4 180C_180L_5 190C_190L_1 190C_190L_2 190C_190L_3 190C_190L_4 190C_190L_5 200C_200L_1 200C_200L_2 200C_200L_3 200C_200L_4 200C_200L_5
1495.37 1538.21 1410.20 1496.29 1504.45 1744.64 1837.11 1745.44 1787.36 1731.63 2214.97 2080.85 2111.28 2101.57 2025.77 2411.76 2441.94 2453.64 2399.02 2428.05 2880.42 2868.06 2804.98 2734.99 2739.35 3131.45 3307.49 3243.72 3160.18 3149.45 3667.40 3668.45 3761.57 3674.25 3641.42 3880.67 4088.36 3955.60 4035.32 4074.69 4495.56 4388.51 4460.18 4431.07 4665.39 4971.06 4935.51 4988.04 5059.23 4963.08
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0007% 0.0029% 0 0 0 0.0003% 0 0.1388% 0 0 0 0 0.0017% 0 0 0 0 0 0 0 0 0 0 0 0 0
Proposed model Time(s) 127.68 12122.99 1658.94 11871.55 4934.80 2961.01 639.22 3092.61 26821.12 9778.38 32778.93 31860.87 39758.12 7939.11 269.34 370.21 38758.95 34077.84 17439.27 31339.56 22678.05 443,200 4 43,200 9520.43 4 43,200 4 43,200 15923.96 41862.29 40925.53 4 43,200 4 43,200 4 43,200 26615.90 4 43,200 28663.62 4 43,200 4 43,200 4 43,200 4 43,200 31272.24 2436.15 4 43,200 37026.93 2765.22 4 43,200 3367.65 4 43,200 4 43,200 4 43,200 4 43,200
Objective RPD
Time I (s)
Time II (s)
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
210.48 233.76 391.54 281.88 213.67 317.61 282.28 314.46 321.66 416.07 2011.95 502.57 441.59 457.27 447.42 534.20 686.70 892.64 665.12 667.84 896.97 933.08 832.61 805.40 908.65 3166.34 1154.51 1743.27 1124.00 1130.34 1614.98 1373.49 1218.71 1512.30 2125.96 2182.87 2211.79 2255.13 1893.83 1944.22 2898.67 2394.14 2611.70 2565.92 3402.05 3032.81 3637.06 2993.43 3223.69 3388.23
41.75 46.61 39.30 35.97 45.82 53.09 62.96 61.26 50.87 72.33 176.32 80.17 93.07 74.79 94.99 176.04 219.93 204.22 234.34 108.33 273.58 334.01 290.24 292.55 210.27 288.68 351.98 261.35 275.24 272.40 533.55 407.80 423.63 470.40 522.91 452.75 545.88 758.31 501.91 624.89 1052.77 637.40 618.03 622.54 890.13 848.68 739.49 677.92 1158.42 932.53
Total Time 252.23 280.38 430.83 317.85 259.49 370.69 345.23 375.72 372.53 488.40 2188.26 582.73 534.66 532.06 542.41 710.24 906.63 1096.86 899.46 776.17 1170.54 1267.09 1122.85 1097.95 1118.93 3455.02 1506.49 2004.62 1399.24 1402.74 2148.53 1781.29 1642.33 1982.70 2648.87 2635.63 2757.67 3013.45 2395.73 2569.11 3951.44 3031.53 3229.73 3188.45 4292.18 3881.49 4376.55 3671.35 4382.11 4320.76
118
C.-C. Lin, S.-W. Lin / Computers & Operations Research 67 (2016) 113–119
The equality between Model LCL and the proposed model has been proved. The proposed model may now be used to determine the optimal terminal set, which should be identical to that found using Model LCL, except that exactly how the goods are transported from i to j, namely, the value of xkm ij , is unavailable. xkm ij
can be obtained using Model LCL after the Nevertheless, terminals have been determined, if the decision-maker needs to know them. This constitutes a two-stage optimization approach for the ITLP. The difficulty of solving the ITLP now lies in the first stage, which is a mixed 0–1 programming problem. The terminals being determined, the second stage is simply a typical transshipment problem without 0–1 variables. With fewer decision variables, the proposed model is expected to be more efficient than Model LCL. The number of decision variables in Model LCL is j Ij 2 j K j 2 þ j Ij 2 þ j K j while that of the proposed model is only j Ij j I þ K j 2 þ j K j . For example, when
j Ij ¼ j K j ¼ 100, Model LCL would use 25 times more variables than those used by the proposed model.
4. Heuristic The above propositions have shown that with fewer decision variables, the proposed model can find the optimal terminals as those found by using the model of Lin et al. [18]. In addition to proposing a revised model for the ITLP, they also proposed two matheuristics, which have been shown to be capable of giving near optimal solutions with less computation time. Among the two matheuristics, the first matheuristic (Matheuristic-1) performs better. Fig. 1 shows the algorithm of Matheuristic-1. After solving the relaxed Model LCL, Matheuristic-1 sets the yk of all the partially utilized terminals to be 0–1 variables and solves the model again. A drawback of this matheuristic is that the number of
Table 2 Computation results by different matheuristics. Problem
110C_110L_1 110C_110L_2 110C_110L_3 110C_110L_4 110C_110L_5 120C_120L_1 120C_120L_2 120C_120L_3 120C_120L_4 120C_120L_5 130C_130L_1 130C_130L_2 130C_130L_3 130C_130L_4 130C_130L_5 140C_140L_1 140C_140L_2 140C_140L_3 140C_140L_4 140C_140L_5 150C_150L_1 150C_150L_2 150C_150L_3 150C_150L_4 150C_150L_5 160C_160L_1 160C_160L_2 160C_160L_3 160C_160L_4 160C_160L_5 170C_170L_1 170C_170L_2 170C_170L_3 170C_170L_4 170C_170L_5 180C_180L_1 180C_180L_2 180C_180L_3 180C_180L_4 180C_180L_5 190C_190L_1 190C_190L_2 190C_190L_3 190C_190L_4 190C_190L_5 200C_200L_1 200C_200L_2 200C_200L_3 200C_200L_4 200C_200L_5
Optimal objective value (107 )
1495.37 1538.21 1410.20 1496.29 1504.45 1744.64 1837.11 1745.44 1787.36 1731.63 2214.97 2080.85 2111.28 2101.57 2025.77 2411.76 2441.94 2453.64 2399.02 2428.05 2880.42 2868.06 2804.98 2734.99 2739.35 3131.45 3307.49 3243.72 3160.18 3149.45 3667.40 3668.45 3761.57 3674.25 3641.42 3880.67 4088.36 3955.60 4035.32 4074.69 4495.56 4388.51 4460.18 4431.07 4665.39 4971.06 4935.51 4988.04 5059.23 4963.08
Matheuristic-1
Proposed matheuristic
Objective RPD
Time I (s)
Time II (s)
Total Time
Objective RPD
Time I (s)
Time II (s)
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0003% 0 0 0 0 0 0 0 0 0.0003% 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
83.21 275.43 573.00 248.34 188.05 521.92 319.90 365.52 362.11 789.78 1121.15 589.14 480.03 1060.94 176.18 266.42 667.13 789.27 1065.85 706.82 846.49 972.87 919.96 900.39 1076.57 1931.74 1255.20 1867.51 1397.43 7286.20 2962.16 1473.18 1458.86 1507.79 1620.72 1860.17 2183.88 2022.31 1699.72 1704.81 801.92 2499.79 4409.09 2176.28 2234.21 854.59 2750.72 2545.20 2985.47 3096.03
50.47 47.11 44.22 37.26 47.55 46.79 74.02 63.58 51.80 59.90 170.24 77.52 95.72 86.37 99.37 193.52 226.80 187.89 231.21 110.37 279.20 321.40 290.30 298.59 216.40 307.47 351.03 311.04 285.52 282.76 532.98 408.17 434.28 499.60 522.33 473.37 542.83 596.12 423.72 685.78 784.49 807.76 623.26 632.26 693.40 1108.88 753.70 689.38 1242.18 1166.1650
133.68 322.54 617.22 285.60 235.60 568.71 393.92 429.10 413.90 849.67 1291.40 666.66 575.75 1147.31 275.55 459.94 893.93 977.16 1297.06 817.19 1125.69 1294.27 1210.25 1198.98 1292.97 2239.21 1606.23 2178.56 1682.95 7568.96 3495.14 1881.35 1893.14 2007.39 2143.05 2333.54 2726.71 2618.43 2123.44 2390.59 1586.41 3307.55 5032.35 2808.54 2927.60 1963.47 3504.42 3234.59 4227.65 4262.19
0 0 0 0 0.0007% 0.0017% 0 0 0 0 0 0.0005% 0.0014% 0 0 0 0 0.0004% 0 0 0 0 0 0 0 0 0.0003% 0 0 0 0 0 0 0 0 0.0003% 0 0.0003% 0 0 0 0 0 0.0002% 0 0 0 0.0004% 0.0006% 0.0002%
117.64 59.18 87.77 100.17 102.45 69.62 70.41 139.06 88.69 77.61 240.58 215.81 197.47 210.09 230.56 314.66 324.36 343.42 311.83 288.39 398.55 343.82 382.47 431.01 423.58 500.82 602.83 499.62 549.17 518.27 310.70 644.10 719.70 650.02 617.31 870.71 761.08 832.34 839.98 793.73 847.70 997.30 897.19 837.65 886.83 1418.04 1073.61 1149.78 1270.39 1182.37
47.43 46.41 37.39 36.88 46.86 54.44 70.18 68.35 48.08 67.48 152.62 88.01 107.26 82.34 97.02 179.98 220.08 172.44 243.73 109.08 275.04 331.98 281.26 304.55 212.74 332.44 356.23 301.37 298.18 273.68 529.90 522.20 480.24 494.38 465.11 586.11 544.36 582.40 404.45 567.96 792.26 938.21 628.93 589.35 712.35 1157.62 750.70 848.25 1094.56 1066.16
Total Time 165.07 105.58 125.16 137.06 149.30 124.07 140.60 207.40 136.76 145.09 393.20 303.82 304.73 292.43 327.58 494.64 544.44 515.86 555.55 397.47 673.59 675.81 663.73 735.56 636.32 833.26 959.06 800.99 847.35 791.95 840.59 1166.30 1199.94 1144.40 1082.41 1456.81 1305.44 1414.74 1244.43 1361.69 1639.96 1935.51 1526.12 1427.00 1599.18 2575.67 1824.31 1998.03 2364.94 2248.53
C.-C. Lin, S.-W. Lin / Computers & Operations Research 67 (2016) 113–119
required iterations that solve a series of mixed 0–1 programming models to reach the final solution is unpredictable. A long computation time might be required as the problem is complex. One possible approach to controlling the computation time of the unpredictable Matheuristic-1 is to set all of the remaining yk with a value less than 1 to be 0–1 variables after the initial relaxed model is solved. In this way, iterative solving of 0–1 programming models is avoided. Meanwhile, all direct flows between customers and those through terminals with yk ¼ 1 are fixed when solving the following 0–1 programming model. Fig. 2 states the algorithm of the proposed matheuristic. The two matheuristics will be compared in the following section.
5. Computational results This section compares the proposed model with that of Lin et al. [18], as well as the proposed matheuristic with their Matheuristic-1. The test problems comprise only the larger problems in [18] because the smaller problems that range from 10–100 customers/terminals appeared to be too easy to effectively distinguish between the models. The details of generating the test problems can be found in [18]. The test problems were solved using Gurobi 5.5 on a workstation with two 2.6 GHz Intel Xeon E52670 CPUs and 96 GB RAM, which is the same as in [18]. The allowable computation time was also set to 12 h. Table 1 summarizes the computational results by the models. The solution quality is measured using relative percentage deviation (RPD) with respect to the optimal objective value. Time I and II represent respectively the times of Stage I and II. On average, the proposed model used about only 5% of the computation time used by Model LCL, and all of the test problems were solved within the time limit. Although Model LCL has found the optimal solutions to some of the large problems before the time was out, the optimality for these problems was undetermined. The results demonstrate that the proposed model with largely reduced decision variables is significantly more efficient than existing models for the ITLP. Less computation time is required when selecting intermodal terminals, as mentioned above, is the only concern of the decisionmaker. If this is the case, the computation time can be further reduced by 20% because Time II is roughly 25% of Time I. Table 2 compares the proposed matheuristic with Matheuristic-1, both based on the proposed model. The proposed matheuristic uses approximately 50% of the computation time used by Matheuristic-1. A paired t-test suggests that the computation time difference is statistically significant. Though the proposed matheuristic appears more efficient than Matheuristic-1, the improvement is not as significant as that achieved by the proposed model, indicating that the use of matheuristics seems to have reached a limit. Nevertheless, the proposed matheuristic is still helpful. Comparing the results in Tables 1 and 2 reveals that Matheuristic-1 is very unlikely to be helpful in reducing computation time. The computation time of Matheuristic-1 is even slightly longer than that of the proposed model. This suggests that Matheuristic-1 is obsolete after the new model is proposed. As mentioned earlier, this obsolescence may result from the repeated solving of a long series of 0–1 programming problems. Although the proposed matheuristic is slightly less sophisticated than Matheuristic-1 when it comes to optimality, the objective RPD remains at a very low level.
6. Conclusions Multimodal transportation has recently attracted much attention. Among related problems, the intermodal terminal location problem (ITLP) aims to determine the locations of intermodal terminals.
119
According to Lin et al. [18], the ITLP represents a more realistic formulation than others based on hub location problems. Sörensen et al. [17] identified this problem and provided a mixed 0–1 programming model for it. Though Lin et al. [18] have revised the model to make it more efficient, the potential for further refinement remains. This study proposes selecting intermodal terminals first and then routing the flows between customers. In the first stage, exact flows between customer pairs are aggregated into path flows, and decision variables can be largely reduced, accordingly. After the terminal nodes are selected, in the second stage there remains only a simple linear programming problem. The second stage may be omitted if the decision-maker's only concern is the terminal locations. Experimental results show that the improvement via this approach is significant. The computation time can be shortened by around 95%. However, when attempting to further reduce the computation time by using the Matheuristic-1 of Lin et al. [18], no significant improvement can be observed. Therefore, a new matheuristic was devised, in order to reduce the computation time while maintaining the solution quality. The new matheuristic did reduce the computation time by half, but the improvement was not very significant. This fact indicates that the refinement of programming model, and thus the capability of matheuristics, has reached a limit. To more rapidly obtain the solution, other heuristics or metaheuristics might offer better odds.
Acknowledgment The authors thank the Ministry of Science and Technology, Republic of China (Taiwan) for financially supporting this study under Grant no. MOST103-2410-H-182-006.
References [1] European Conference of Ministers of Transport. Terminology on combined transport. 〈http:/www.cordis.lu/transport〉; 1993. [2] Chang T-S. Best routes selection in international intermodal networks. Comput Oper Res 2008;35:2877–91. [3] Hanssen T-ES, Mathisen TA, Jørgensen F. Generalized transport costs in intermodal freight transport. Proced Social Behav Sci 2012;54:189–200. [4] Janic M. Modelling the full costs of an intermodal and road freight transport network. Transp Res Part D 2007;12:33–44. [5] Caris A, Macharis C, Janssens GK. Decision support in intermodal transport: a new research agenda. Comput Ind 2013;64:105–12. [6] Macharis C, Bontekoning YM. Opportunities for OR in intermodal freight transport research: a review. Eur J Oper Res 2004;153:400–16. [7] Bektas T, Crainic T. A brief review of intermodal transportation. In: Taylor G, editor. Logistics engineering handbook. London: CRC Press; 2007. p. 1–16. [8] SteadieSeifi M, Dellaert NP, Nuijten T, Van Woensel T, Raoufi R. Multimodal fright transportation planning: a literature review. Eur J Oper Res 2014;233:1–15. [9] Alumur S, Kara BY. Network hub location problems: the state of the art. Eur J Oper Res 2008;190:1–21. [10] Santos BF, Limbourg S, Carreira JS. The impact of transport policies on railroad intermodal freight competitiveness – the case of Belgium. Transp Res Part D 2015;34:230–44. [11] Farahani RZ, Hekmatfar M, Arabani AB, Nikbakhsh E. Hub location problems: a review of models, classification, solution techniques, and applications. Comput Ind Eng 2013;64:1096–109. [12] Arnold P, Peeters D, Thomas I, Marchand H. Pour une localisation optimale des centre de transbordement intermodaux entre réseaux de transport: formulation et extensions. Can Geogr 2001;45:427–36. [13] Arnold P, Peeters D, Thomas I. Modelling a rail/road intermodal transportation system. Transp Res Part E 2004;40:255–70. [14] Racunica I, Wynter L. Optimal location of intermodal freight hubs. Transp Res Part B 2005;39:453–77. [15] Limbourg S, Jourquin B. Optimal rail-road container terminal locations on the European network. Transp Res Part E 2009;45:551–63. [16] Ishfaq R, Sox CR. Intermodal logistics: the interplay of financial, operational and service issues. Transp Res Part E 2010;46:926–49. [17] Sörensen K, Vanovermeire C, Busschaert S. Efficient metaheuristics to solve the intermodal terminal location problem. Comput Oper Res 2012;39:2079–90. [18] Lin CC, Chiang YI, Lin SW. Efficient model and heuristics for the intermodal terminal location problem. Comput Oper Res 2014;51:41–51.