- Email: [email protected]
An efficient two-phase exact algorithm for the automated truck freight transportation problem
Accepted Manuscript An efficient two-phase exact algorithm for the automated truck freight transportation problem Peng Wu, Feng Chu, Ada Che, Yunfei Fang PII: DOI: Reference:
S0360-8352(17)30172-9 http://dx.doi.org/10.1016/j.cie.2017.04.030 CAIE 4717
To appear in:
Computers & Industrial Engineering
Received Date: Revised Date: Accepted Date:
22 September 2016 25 February 2017 14 April 2017
Please cite this article as: Wu, P., Chu, F., Che, A., Fang, Y., An efficient two-phase exact algorithm for the automated truck freight transportation problem, Computers & Industrial Engineering (2017), doi: http://dx.doi.org/10.1016/ j.cie.2017.04.030
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
An efficient two-phase exact algorithm for the automated truck freight transportation problem Peng Wua,b , Feng Chub,c,∗, Ada Chea , Yunfei Fangd a
School of Management, Northwestern Polytechnical University, 710072 Xi’an, China b Laboratory IBISC, University of Evry-Val d’Essonne, 91020 Evry, France c Management Engineering Research Center, Xihua University, 610039 Chengdu, China d School of Economics & Management, Fuzhou University, 350116 Fuzhou, China
Abstract Fang et al. (2013) have developed an integer linear program and an exact algorithm for the automated truck transportation freight problem with lane reservation. However, due to its NP-hard nature, their proposed method becomes difficult to solve large-size problems within acceptable time. In this paper, we firstly present an improved integer linear program by adding valid inequalities and identify that its several special cases are classical combinatorial optimization problems. Based on analyzed properties, a new efficient two-phase exact algorithm is developed. Computational results on benchmark and new larger-size instances with up to 700 nodes and 55 tasks show that the new algorithm outperforms very favorably the state-of-the-art ones. Keywords: transportation, automated truck, lane reservation, integer programming, exact algorithm 1. Introduction Efficient shipments of cargos has attracted much attention and considerable freight transportation planning problems have been investigated extensively over the past decades (Prins, 2004; Goksal et al., 2013; Shaabani and Kamalabadi, 2016). However, increasing travel demand results in increasingly severe congestion, which causes many problems in transportation, such as low efficiency, unpredictable transport time, traffic accidents, and fuel ∗
Corresponding authors: [email protected] (F. Chu)
Preprint submitted to Computers & Industrial Engineering
May 13, 2017
waste. These problems increasingly prevent the freight transportation from being operated in an efficient, reliable and safe fashion (Fang et al., 2013). Introducing automated driving for trucks would be a promising solution to cope with such challenge, as automated trucks could provide remarkable advantages such as high safety and efficiency, and lower fuel consumption. Unlike manually driven trucks, automated ones must have the ability of detecting possible dangers and responding to them correctly and promptly. Dedicated truck lanes would be ideal in this sense. Since constructing new network dedicated to automated trucks may be infeasible due to the high costs and limited geographic space, converting existing general-purpose (GP) lanes in the existing network to dedicated ones is an effective alternative. But due to the exclusive use of reserved lanes by automated trucks, the available lanes in the network for GP vehicles are reduced, and negative impact, such as the increase in travel time of GP vehicles, will be generated on the adjacent lanes. It is necessary to well decide appropriate lanes to be reserved to achieve the safe and time-guaranteed automated truck transportation, while minimizing the negative traffic impact. Such an optimization problem is called the automated truck transportation problem with lane reservation (ATP) (Fang et al., 2013). We note that there have also been studies investigating lane reservation for other applications, such as large sport events, hazardous material transportation, bus transit (Wu et al., 2009; Zhou et al., 2013; Che et al., 2015; Wu et al., 2015, 2016). Fang et al. (2013) have formulated an integer linear program (ILP) and developed an exact cut-and-solve algorithm for the ATP. However, due to the NP-hardness of the ATP, their proposed methods become difficult to solve large-size problems within acceptable computational time. In this paper, we first provide an improved ILP by adding valid inequalities. Then, we identify that several special cases of the ATP are classical combinatorial optimization problems. Based on the analyzed properties, a new efficient two-phase exact algorithm is developed. Computational results on 120 benchmark and 210 new larger-size instances with up to 700 nodes and 55 tasks confirm the effectiveness of the proposed algorithm. The remainder of the paper is organized as follows. Section 2, recalls the problem description and provides the improved ILP. In Section 3, we derive several optimal properties of the ATP. Based on them, a new efficient exact algorithm is presented in Section 4. Section 5 reports the computational results. Section 6 concludes this study.
2
2. Problem description and formulation The ATP considered in this study has been addressed by Fang et al. (2013). For the sake of self-consistency, the problem description is first recalled as follows. The ATP can be defined on a transportation network that can be represented by a directed graph G(N, A) with a node set N and an arc set A. A node (resp. an arc) represents a road intersection (resp. a road segment). Given a set of automated truck transportation tasks to be accomplished and their corresponding origin-destination (OD) pair, the ATP consists of optimally selecting lanes to be reserved in the existing network and designing a reserved path for each task in order to ensure that it can be completed within its travel deadline safely. However, such lane reservation reduces the available lanes of GP vehicles such that the negative impact such as the increase of travel time on the adjacent lanes may be caused. The objective is to minimize the total negative impact caused by reserved lanes. As stated in Fang et al. (2013), some assumptions are made to facilitate the formulation of the ATP. First, the negative impact caused by a reserved lane is assumed as the increase of travel times on the remaining GP lanes. More details on the impact parameters can be found in Section 5. Second, at most one reserved lane is allowed on each road segment. Third, there is only one path for each task from its origin to destination in order to ensure the transport safety and the path only consists of reserved lanes. Fourth, there are at least two lanes on each road segment allowing one reserved lane. We first summarize the parameters and decision variables in Table 1.
3
Table 1: Notation and its explanation for the formulation
Notation Explanation G = {V, A} transportation network, where N and A denote the set of nodes and arcs, respectively i, j node index, i, j ∈ N k task index, k ∈ K O set of transportation tasks with |K| tasks D set of destination nodes, D ⊆ N ok origin node of task k ∈ K, ok ∈ O dk destination node of task k ∈ K, dk ∈ D Tk travel deadline of task k ∈ K Tij travel time on a reserved lane on arc (i, j) ∈ A Cij negative impact caused by a reserved lane on arc (i, j) ∈ A Zij equal to 1 if arc (i, j) is reserved, and 0 otherwise, (i, j) ∈ A Xkij equal to 1 the path of task k pass arc (i, j) which is reserved, and 0 otherwise, (i, j) ∈ A, k ∈ K 2.1. Problem formulation Before giving an improved model, we first recall the existing ILP for the ATP proposed by Fang et al. (2013), shown as follows. X IP : min Cij Zij (1) (i,j)∈A
X
s.t.
Xkok i = 1, for all k ∈ K
(2)
(ok ,i)∈A
X
Xkidk = 1, for all k ∈ K
(3)
(i,dk )∈A
X
Xkij =
j:(i,j)∈A
X
X
Xkji , for all j ∈ N \{ok , dk }, and all k ∈ K (4)
j:(j,i)∈A
Xkij Tij ≤ Tk , for all k ∈ K
(5)
(i,j)∈A
Xkij ≤ Zij , for all (i, j) ∈ A, and all k ∈ K Zij ∈ {0, 1}, for all (i, j) ∈ A Xkij ∈ {0, 1}, for all (i, j) ∈ A and all k ∈ K
4
(6) (7) (8)
Objective (1) is to minimize the total negative impact of all reserved lanes. Constraints (2)-(4) guarantee that there exists a feasible path for each OD pair. To be more specific, constraint (2) (resp. (3)) implies that there exists only one arc outgoing from (resp. coming into) origin node ok (resp. destination node dk ). Constraint (4) ensures the flow conservation for intermediate nodes between origin and destination for each task k ∈ K. Constraint (5) indicates that the total travel duration for task k from its origin to destination should not exceed its travel deadline. Constraint (6) ensures that task k can pass a reserved lane on arc (i, j) ∈ A only if this arc is reserved. Constraints (7) and (8) enforce the bounds of all decision variables. The benefit to be achieved for the ATP is the time-efficient automated truck transport service, i.e., each task should be completed within the given transportation deadline. Based on this goal, constraint (5) is formulated to specify the requirement of time-efficient transportation. For the ATP, the transport delay is not accepted, but trucks are allowed to arrive early since they achieve higher transport efficiency. We note that the model IP can be adapted to the case that does not allow trucks arrive too early but allows an acceptable delay by changing constraint (5) to a time window constraint. 2.2. An improved integer linear program for the ATP We improve the above formulation based on the following observations. Observation 1. For any task k ∈ K, there will be no arcs entering into (resp. outgoing from) its origin ok (resp. destination dk ) on its path. With Observation 1, we add the following valid inequalities into IP without excluding optimal solutions. Xkiok = 0, for all (i, ok ) ∈ A and all k ∈ K Xkdk i = 0, for all (dk , i) ∈ A and all k ∈ K
(9) (10)
Obviously, constraints (9) and (10) reduce the search space of the original problem since part of variables are prefixed. Observation 2. For any task k ∈ K, each node in the network will be passed at most once. Note that if a node in the network is passed more than once by a task (i.e., cycles exist on the transport path), this obviously generates larger negative 5
impact compared with the case without cycles. With Observation 2, we also add the following constraints into IP . X Xkij ≤ 1, for all j ∈ N \{ok , dk } and all k ∈ K (11) j:(i,j)∈A
X
Xkji ≤ 1, for all j ∈ N \{ok , dk } and all k ∈ K
(12)
j:(j,i)∈A
Constraints (11) and (12) are also valid inequalities, which tighten the search space of the original problem. With the newly obtained constraints, we derive the following improved IP model. X IP 0 : min Cij Zij (i,j)∈A
s.t. s.t. Constraints (2)-(12) P + Remark 1. The improved formulation employs k (|A− k |+|Ak |)+2|K|(|N |− 2) more constraints as compared with Fang et al. (2013)’s formulation, where + |A− k |(resp. |Ak | denotes the number of arcs entering into origin ok (resp. outgoing from destination dk ). P + Remark 2. The prefixing of k (|A− k | + |Ak |) variables and relatively more constraints help reduce search space. Computational results in Section 5.1 show that IP’ can save 19.11% average time compared with IP. Fang et al. (2013) stated that the ATP is NP-hard based on their observation but the proof was not given. The detailed proof is given below. Theorem 1. The ATP is NP-hard. Proof. The proof is based on the fact that constrained shortest path problem (CSPP) reduces to the special case of the ATP with only one task. Let us consider the following instance of CSPP. Given a graph G0 = (N 0 , A0 ) associated with two weights cij > 0 (say, cost) and dij > 0 (say, delay), (i, j) ∈ A0 , two distinguished nodes s and t, s, t ∈ N 0 and a positive value T . The CSPP consists of finding a minimum cost s-t path with its total delay being equal to or less than T . We then show how to transform the above CSPP instance into an instance of the special case of the ATP with only one task. Denote the only task as 6
task 1. Let us map (N 0 , A0 ) to (N, A) (i.e., N 0 and A0 are mapped to N and A, respectively), s to o1 and t to d1 . Cij , Tij , and T1 are equal to cij , dij , and T , respectively. Through such linear transformation, the CSPP is reduced into the special case of the ATP with only one task. As the CSPP is NP-hard even for acyclic networks (Xiao et al., 2005), the ATP with one task is consequently NP-hard. Certainly, the ATP in general case is NP-hard. 3. Property analysis for the ATP In this section, we first investigate several special cases for the ATP. Note that these special cases correspond to classical combinatorial optimization problems and can be tackled using existing techniques. The potential benefits are that if an instance is recognized as one special case of them, then it can be efficiently solved accordingly. Then, the ATP in the general case is analyzed. 3.1. Special cases of the ATP Case 1: The ATP with only one task and large task travel deadline. When the travel deadline of the task is large enough, then the travel deadline constraint can be relaxed. Obviously, the special ATP in Case 1 is equivalent to find a reserved path with minimum impact. Then, the following proposition is straightforward. Proposition 1. The special ATP in Case 1 is equivalent to a shortest path problem. Remark 3. The special ATP in Case 1 is polynomial solvable as the shortest path problem that can be efficiently solved by the Dijkstra shortest path algorithm (Dijkstra, 1959) whose time complexity is O(|N |2 ). Case 2: The single-source ATP with large task travel deadline. For this case, we have the following proposition. Proposition 2. The special ATP in Case 2 is equivalent to a directed Steiner tree problem (DSTP).
7
Proof. Let us consider an instance of DSTP as follows. Given a directed graph G = (V, E) with positive weights on the edges, a set of terminal vertices V 0 ⊆ V , a root node vertex r. i, j ∈ V , edge (i, j) ∈ E and its weight is denoted by wij . The instance of DSTP consists of finding a minimum weight out-branching tree T rooted at r, such that the vertices in V 0 are included in T . We now show how to transform the DSTP instance into an instance of the special ATP in Case 2. Let us map graph G = (V, E) to graph G = (N, A) (i.e., V and E are mapped to N and A, respectively), V 0 to D and r to the only origin node. The weight Cij is equal to wij . Then, designing the task paths with minimal impact of lane reservation is equivalent to finding a minimum weight out-branching tree T . With such linear transformation, the DSTP is reduced into the special Case 2 of ATP. Remark 4. The special ATP in Case 2 is NP-hard, as the DSTP is known to be NP-hard (Karp, 1972). Case 3: The ATP with only one task. By the proof of Theorem 1, we have the following proposition. Proposition 3. The special ATP in Case 3 is equivalent to a constrained shortest path problem. Remark 5. The special ATP in Case 3 is NP-hard. 3.2. The ATP in general case Generally, an ATP contains multiple tasks and different tasks are allowed to share reserved lanes. Since the path design of any task may be influenced by other tasks’ paths due to the requirement of minimizing negative impact of reserved lanes. Consequently, the path of any task may be not an optimal path determined by solving its corresponding constrained shortest path problem. That is to say, solving an ATP with |K|(|K| > 1) tasks may be not equivalent to independently solving |K| constrained shortest path problems. For the ATP in general case, the following proposition is straightforward. Proposition 4. Solving an ATP with multiple tasks is equivalent to finding an optimal loopless path respecting travel deadline constraint for each task to form the best path combination such that the total negative impact of reserved lanes is minimized. 8
As previously analyzed, we know that the ATP with only one task is NPhard even for acyclic networks. This means that the ATP with multiple tasks in the network with cycles is even harder to handle. In the following section, an exact method to efficiently solve the ATP in general case is developed. 4. Two-phase exact algorithm for the ATP For an ATP in general case, Fang et al. (2013) proposed a cut-and-solve algorithm, which can solve problem instances with up to 150 nodes in the network and 30 tasks within 18000 CPU seconds. In this paper, a new efficient two-phase exact algorithm is developed to efficiently solve the largersize ATP. The algorithm is composed of two major phases. In the first phase, all feasible paths respecting the travel deadline constraint are enumerated for each task k ∈ K. An optimal lane reservation scheme and task paths are then determined in the second phase. We detail our new optimal algorithm in what follows. 4.1. Phase 1: Task path enumeration As indicated by Proposition 3, the ATP with one task is a constrained shortest path problem that consists of finding a loopless reserved path respecting the travel deadline constraint with minimal negative impact. Thus, for any task k ∈ K, let Pk denote the set of all loopless paths connecting its origin and destination and respecting its travel deadline (i.e., the total travel duration is equal to or less than Tk ). Remark 6. The optimal path of task k ∈ K of the ATP must be in set Pk . It is not hard to find that the problem determining the set Pk for each task k ∈ K is equivalent to finding all loopless paths with their travel duration being equal to or less than Tk in the direct graph G(N, A). We note that such problem can be efficiently solved using the well-known Yen’s K-shortest loopless path algorithm (Yen, 1971) with its time complexity O(K|N |(|A| + |N |log|N |) (Bouillet, 2007), which belongs to a kind of deviation algorithms ranking the first K loopless paths for given pair of nodes. To more efficiently obtain the set Pk , ∀k ∈ K, the search space for each k ∈ K is reduced before using Yen’s K-shortest loopless path algorithm. For k ∈ K, a ∈ A, we define the possibly passed arc set Ak as follows. Ak = {(i, j)|ϕ(ok , i) + Tij + ϕ(j, dk ) ≤ Tk }, k ∈ K 9
(13)
where ϕ(ok , i)(resp. ϕ(j, dk ) denotes the shortest path from ok to i(resp. j to dk ) when all the arcs in the network are reserved. Note that arcs belonging to A\Ak would not be passed by task k, otherwise the travel deadline constraint will be violated. In other words, set Ak includes all arcs in the network that may be passed by task k ∈ K. Since set Ak is a subset of A, obviously the resolution of the path enumeration for task k will be accelerated. Remark 7. From Proposition 4, we can observe that the path selected for task k ∈ K in the optimal solution of the ATP may not be the travel deadline constrained path with minimal negative impact. As the above remark, after obtaining the set Pk for each k ∈ K, the path for any task cannot be simply determined to be the one in the set Pk with the minimal negative impact. According to Proposition 4, the ATP is to find the best task path combination so as to minimize the total impact. 4.2. Phase 2: Lane reservation and task path determination The candidate path sets Pk , ∀k ∈ K are determined in the above phase. In the second phase, we propose an integer programming method to determine the optimal lane reservation scheme and task path from Pk for each task k ∈ K. To formulate the model of determining optimal lane reservation and k all task paths, we need to additionally define parameter δpij , if path p ∈ Pk k passes arc (i, j) ∈ Ak , δpij = 1, and 0 otherwise, and a new variable Ypk below. Ypk : = 1 if the path p ∈ Pk is selected; and 0 otherwise, ∀k ∈ K. Then, the new ILP for determining optimal lane reservation and task path is given as follows. X IP 00 : min Cij Zij (i,j)∈A
X
s.t.
k δpij Ypk
≤ Zij , for all k ∈ K, and all (i, j) ∈ A
(14)
p∈Pk
X
k Ypk ≤ Zij , for all k ∈ K, and all (i, j) ∈ A δpij
(15)
p∈Pk
Ypk , Zij ∈ {0, 1}, ∀p ∈ Pk , for all k ∈ K, and all (i, j) ∈ A
(16)
where constraint (14) indicates that only one path is selected from the candidate path set Pk . Constraint (15) ensures that the path of task k, ∀k ∈ K 10
involving arc (i, j), ∀(i, j) ∈ A only if this arc is reserved. Constraint (16) gives the ranges of decision variables. Note that IP 00 is an integer linear program which can be tackled by the optimization software such as CPLEX. Algorithm 1 Two-phase exact algorithm for the ATP 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11:
initialize Pk = ∅ for all k ∈ K, k=0; while k ≤ |K|, do: let l = 1; for task k, define set Ak based on formula (13); for OD pair (ok , dk ), compute the lth shortest path using Yen’s K-shortest loopless path algorithm, and record the travel duration dkl and its path pkl ; if dkl ≤ Tk , Pk = Pk ∪ {pkl }, l = l + 1 and goto step 5; otherwise k = k + 1 and goto step 2; end while construct model IP 00 using sets Pk , ∀k ∈ K obtained above; solve IP 00 exactly using CPLEX IP solver; output the lane reservation scheme and the corresponding paths for tasks.
4.3. Overall algorithm for the ATP The overall algorithm for the ATP can be outlined in Algorithm 1, in which steps 1-8 (i.e., Phase 1) aim to enumerate all feasible paths for each task and steps 9-11 are to determine the final lane reservation scheme and task paths (i.e., Phase 2). 5. Computational experiments In this section, we conduct numerical computational experiments to show the performance of the proposed algorithm. Our algorithm is coded in C++ language and combined with Yen’s K-shortest loopless path algorithm (Yen, 1971) and CPLEX (12.6) IP solver with default settings. All the experiments are done on a PC with 2.5 GHz and 2.95 GB RAM with windows 7 system. The performance of the proposed algorithm is evaluated on eighty-one groups of instances with five instances each group, including 120 benchmark instances (Fang et al., 2013) and 210 new larger-size instances. These new instances are randomly generated based on the way in Wu et al. (2009) and Fang et al. (2013). Waxman’s network model Waxman (1988) is used to 11
generate the network G(N, A). D = 2|A|/|N | is called average node degree. To be more specific, the nodes of G(N, A) are randomly distributed in a square area [0, 100] × [0, 100], the existence of arc (i, j) between nodes i and j is dependent by a probability function a exp (−Lij /bLmax ), where Lij and L are the Euclidean distance between nodes i and j and the maximum Euclidean distance between any pair of nodes, respectively, and 0 ≤ a ≤ 1 and 0 ≤ b ≤ 1. The origin and destination nodes ok and dk , k ∈ K are randomly generated from set N . Tij is computed by Lij /Vij , where Vij denotes the average travel speed on a reserved lane of arc (i, j) and is set as 60 km/h. Let Tij0 denote the travel time on arc (i, j) without reserved lanes, which is approximately computed by the widely used BPR function, defined as Tij0 = Tij (1 + α(fij /capij )β )), where fij and capij are the traffic flow per unit time within the considered time horizon and capacity of arc (i, j), respectively. According to Wu et al. (2015), fij /capij is randomly generated in [0.5, 1.2] and α = 0.15 and β = 4. The impact of a reserved lane on arc (i, j) is defined as Cij = Tij0 /(Mij −1), where Mij is the total number of lanes on arc (i, j), as is the case in Wu et al. (2009) and Fang et al. (2013). Therefore, the negative impact due to a reserved lane is proportional to the travel time without reserved lanes and inversely proportional to the number of lanes. If no lane is reserved, then Cij =0. As indicated in Princeton and Cohen (2011), an actually statistical result showed that the travel time of the non-reserved lanes increased about 53% after one of the three lanes was reserved in A1 highway in Paris, which is close to the theoretical result (50%) obtained by the above formula. Tk is defined as Lk + λk (L0k − Lk ), where Lk (resp. L0k ) is the shortest travel time from origin ok to destination dk when all arcs in the network are reserved (resp. no lanes in the network are reserved) and λk is a given parameter, which is randomly generated in [0, 1]. For brevity, let Tcp , Tcs and Ttp denote the average computational time (CPU seconds) spent by CPLEX IP solver solving IP 0 (i.e., solving the ATP with a direct use of CPLEX), the CS algorithm (Fang et al., 2013) and by our two-phase algorithm for five instances of each group, respectively. Besides, let Ttp-f (resp. Ttp-s ) denote the average CPU time (seconds) spent by the first (resp. second) phase of our proposed algorithm. Note that the computational times of both methods are limited to 18000s (i.e., five hours), as is the case in Fang et al. (2013).
12
5.1. Model comparison To demonstrate that the proposed improved model (see Section 2.2) is more efficient, we compare it with the model proposed by Fang et al. (2013) (Fang’s model in short) by solving a number of instances. The two models for these instances are both solved by CPLEX. The comparison results are summarized in Table 2. Saved time rate means the reduced time rate by the improved model compared with the one proposed by Fang et al. (2013). We can observe from Table 2 that the computational time of the improved model is less than that of Fang’s model and the former saves an average 19.11% time compared with the latter. This indicates that the improved model IP 0 is more efficient than the model proposed by Fang et al. (2013). Table 2: Performance of the improved model Group 1 2 3 4 5 6 7 8 Average
|N | 60 60 70 70 80 80 90 90
|K| 25 30 25 30 25 30 25 30
D 7 7 7 7 7 7 7 7
Time of Fang’s model (s) 9.95 26.73 40.99 93.11 46.62 124.76 171.08 317.12 103.80
Tcp 6.75 22.89 36.25 64.78 36.19 102.64 99.24 302.96 83.96
Saved time rate(%) 32.09 14.37 11.56 30.43 22.36 17.73 41.99 4.47 19.11
5.2. Results for benchmark instances in Fang et al. (2013) To evaluate the performance of the proposed algorithm, we first compare it with benchmark instances with the state-of-the-art algorithm (i.e., the CS algorithm proposed by Fang et al. (2013)) in terms of computational time in obtaining optimal solutions. The results are reported in Tables 3 and 4. Table 3 reports the results for instances with fixed number of nodes |N | = 100 and average node degree D ranging from five to twelve. It can be observed from Table 3 that the CPU time of the CS algorithm Tcs varies from 1.59s to 8900.67s with its average value being 873.18s, while that of our algorithm Ttp varies from 0.62s to 1080.95s with its average value being 85.36s. Ttp is less than Tcs over all groups 9-23 except the small-size groups 9, 10 and 12. On average, Tcs is eleven times more than Ttp . This indicates that our algorithm is much more efficient than the CS algorithm in terms of computational time. Moreover, Tcs increases sharply with |K| for a given average node degree, 13
Table 3: Comparison results for instances with |N |=100 and various node degrees Group 9 10 11 12 13
|N | 100 100 100 100 100
|K| 10 15 20 25 30
D 5 5 5 5 5
Tcs 1.59 1.56 41.82 37.21 63.45
Ttp-f 3.42 1.52 8.68 57.98 3.80
Ttp-s 0.39 0.35 0.95 7.05 1.09
Ttp 3.80 1.87 9.64 65.03 4.89
Tcs /Ttp 0.42 0.84 4.34 0.57 12.98
14 15 16 17 18
100 100 100 100 100
10 15 20 25 30
7 7 7 7 7
6.41 62.32 315.98 1193.88 1288.72
0.52 2.46 2.87 1.75 1.75
0.09 0.26 0.32 0.24 0.30
0.62 2.72 3.19 1.98 2.05
10.38 22.88 99.10 601.65 627.73
19 20 21 22 23 Average
100 100 100 100 100
10 15 20 25 30
12 12 12 12 12
18.67 78.30 496.89 1590.22 8900.67 939.85
2.45 5.79 67.25 10.61 197.38 24.55
0.22 0.36 16.23 0.77 883.58 60.81
2.67 6.14 83.48 11.38 1080.95 85.36
7.00 12.75 5.95 139.72 8.23 11.01
especially for larger D = 7 and 12, whereas Ttp varies slightly and it does not necessarily increase with |K|. For example, Ttp for groups 8 and 9 decreases from 3.19s to 1.98s when |K| increases from 20 to 25. On the other hand, we observe that Ttp-s are positively correlated with Ttp-f over groups 1-15 (i.e., the larger Ttp-f is, the larger Ttp-s is). For example, Ttp-f ’s are 8.68s and 57.98s for groups 3 and 4, respectively, and Ttp-s ’s are 0.95s and 7.05s, respectively. The main reason is that a larger Ttp-f implies that more task path candidates exist for the tasks, which means more integer variables Ypk in IP 00 in step 5 of our algorithm such that it is more difficult to be solved. Besides, it can be seen that generally Ttp-f is larger than Ttp-s , which shows the first phase consumes more time. An exception is group 23 (i.e., the largest-size group) where Ttp-s is larger than Ttp-f . The reason is that there exist many task path candidates and it makes the model IP 00 formed in second phase much difficult to be solved. Table 4 presents the results when |N | is increased from 110 to 150 and |K| ranges from 10 to 30 for a given average node degree. In Table 4, we can see that Tcs varies from 6.27s to 1878.07s with its average value being 788.46, whereas Ttp ranges from 5.70s to 108.31s with it average value being 21.01s. Ttp is less than Tcs over groups 24-32, except the smallest-size group 14
Table 4: Comparison results for instances with fixed node degree and |N |= 110-150 Group 24 25 26 27 28 29 30 31 32 Average
|N | 110 110 120 120 130 130 140 140 150
|K| 10 15 15 20 20 25 25 30 30
D 7 7 7 7 7 7 7 7 7
Tcs 6.27 18.02 103.29 121.68 299.71 1406.91 1575.25 1686.95 1878.07 788.46
Ttp-f 6.07 7.93 16.20 4.56 7.68 9.36 5.19 17.05 100.21 19.36
Ttp-s 0.58 0.85 1.70 0.55 0.59 0.61 0.51 1.31 8.09 1.64
Ttp 6.65 8.78 17.90 5.11 8.28 9.97 5.70 18.36 108.31 21.01
Tcs /Ttp 0.94 2.05 5.77 23.81 36.21 141.09 276.35 91.89 17.34 37.54
24. The CS algorithm spends more than 37 times average time of that by our algorithm. This further shows the efficiency of our algorithm as compared with the CS algorithm proposed by Fang et al. (2013). 5.3. Results for new larger-size instances In order to further evaluate the performance of the proposed method for solving larger-size problems, 42 new larger-size problem groups are tested. Computational results are reported in Tables 5-7. Table 5: Computational results for instances with |N |= 160-200 Group 33 34 35 36 37 38 39 40 41 42 Avg.
|N | 160 160 170 170 180 180 190 190 200 200
|K| 20 30 30 35 35 40 40 45 45 50
Tcp 889.01 4312.34 13201.26 16334.21 >14273.68
Tcs 368.72 2432.11 6848.21 11329.37 17582.12 >12856.05
Ttp-f 8.79 11.04 2.75 2.89 2.65 2.58 7.13 8.21 5.13 5.82 5.70
Ttp-s 1.02 0.69 0.22 0.24 0.32 0.37 0.86 0.95 0.41 0.45 0.55
Ttp 9.81 11.73 2.97 3.13 2.97 2.95 7.99 9.16 5.54 6.27 6.25
Table 5 presents the results of large-size instances with |N | increasing from 160 to 200, |K| varying from 20 to 50 and D=7. We can observe in Table 5 that the CPU time consumed by our algorithm is far less than those by CPLEX and the CS algorithm over all groups 33-42. As the problem size increases, Tcp and Tcs increase exponentially, whereas Ttp varies slightly. It 15
is worthwhile to note that CPLEX and the CS algorithm can only exactly solve the four (resp. five) of ten groups within 18000s, whereas our algorithm can exactly solves all the groups with an average time 6.25s. Besides, we can observe that Ttp-f is larger than Ttp-s over all the groups 33-42. Table 6: Computational results for instances with |N |= 300-700 Group 43 44 45 46 47 48 49 50 51 Avg. 52
|N | 300 300 400 400 500 500 600 600 700
|K| 30 40 40 45 45 50 50 55 55
D 5 5 5 5 5 5 5 5 5
700
60
5
53 54 55 56 57 58 Avg. 59
300 300 400 400 500 500
30 40 40 45 45 50
10 10 10 10 10 10
600
50
10
Tcp -
Tcs -
Ttp-f 24.02 26.51 32.22 33.73 41.73 65.12 295.27 741.29 1753.30 334.80 -
Ttp-s 1.75 2.27 1.99 1.93 17.27 23.17 454.89 627.30 6699.99 870.06 -
Ttp 25.77 28.78 34.21 35.66 59.00 88.29 750.16 1368.59 8453.29 1204.86 -
-
-
69.11 248.37 210.23 170.02 869.55 765.37 388.78 -
5.99 20.75 16.25 7.57 99.91 85.12 39.27 -
75.10 269.12 226.48 177.59 969.46 850.49 428.04 -
Table 6 reports the computational results of larger-size instances with |N | increasing from 300 to 700, |K| varying from 30 to 60 under two scenarios D=5 and 10. It can be found that CPLEX and the CS algorithm by Fang et al. (2013) cannot generate optimal solutions for any group within 18000s (i.e., they lose their power for these large-size instances totally), whereas the proposed algorithm is able to exactly solve the instances with up to 700 nodes and 55 tasks with average node degree D = 5 and 500 nodes and 50 tasks with average node degree D = 10, respectively. Due to the NP-hardness of the problem, we find that the proposed algorithm rapidly increases with the problem size for each given average node degree. For example, Ttp for group 43 is 25.77s, while for group 51 is 8453.29s. Moreover, it is not hard to find that the increase of Ttp is mainly caused by the increase of Ttp-s . We also observe that our algorithm consumes more time to solve instances with 16
Table 7: Comparison for sensitivity analysis of Cij = rij Tij0 /(Mij − 1) Group 60 61 62 63 64 Avg. 65 66 67 68 69 Avg. 70 71 72 73 74 Avg.
rij [0.5, 1) [0.5, 1) [0.5, 1) [0.5, 1) [0.5, 1)
|N | 200 300 400 500 600
|K| 20 20 20 20 20
1.0 1.0 1.0 1.0 1.0
200 300 400 500 600
20 20 20 20 20
200 300 400 500 600
20 20 20 20 20
(1, (1, (1, (1, (1,
1.5] 1.5] 1.5] 1.5] 1.5]
Ttp-f 3.34 6.11 13.12 17.35 69.99 21.98 2.59 3.88 17.15 30.24 67.44 24.26 3.56 5.22 14.98 17.11 80.01 24.18
Ttp-s 0.27 0.45 0.99 1.93 3.99 1.53 0.32 0.53 1.30 2.41 5.74 2.06 0.32 0.46 0.92 1.31 4.65 1.53
Ttp 3.61 6.56 14.11 19.28 73.98 23.51 2.91 4.41 18.45 32.65 73.18 26.32 3.88 5.68 15.90 18.42 84.66 25.71
a larger average node degree when |N | and |K| are given. For example, Ttp for group 48 is 88.29s, while for group 58 is 850.49s. The possible reason is that a larger average node degree may result in more task path candidates for each task, which requires more computational effort for both phases in the proposed algorithm. We note that the proposed algorithm loses power for solving groups 52 and 59 due to the lack of memory in the first phase. We have also conducted experiments to evaluate the sensitivity of different impact parameters on the performance of the proposed algorithm. Table 7 reports the computational results. Note that the intervals [0.5, 1) and (1, 1.5] are used to generate smaller and larger impact parameters, respectively. It can be seen that the ranges of Ttp under the three scenarios are 3.61-73.98s, 2.91-73.18s, and 3.88-84.66s, respectively. Moreover, Tcs ’s under the three scenarios are 23.51s, 26.32s, and 25.71s, respectively. These results indicate that our algorithm is insensitive to the changes of Cij . 6. Conclusion In this paper, we have revisited the automated truck transportation problem with lane reservation proposed by Fang et al. (2013). For the problem, we first propose valid inequalities for the integer linear program proposed 17
by Fang et al. (2013). Computational comparison results indicate that these valid inequalities are effective in saving computational time. Furthermore, we have investigated several special cases of the considered problem, which can be identified to be classical combinatorial optimization problems. To efficiently solve the problem, we devise a new efficient two-phase exact algorithm based on analyzed properties. Computational results demonstrate that the proposed algorithm significantly outperforms the state-of-the-art algorithm and it can solve larger-size instances with up to 700 nodes and 55 tasks. Future research directions may include the following aspects: (i) one important direction is to develop more efficient techniques for the resolution of IP 00 to further accelerate the proposed algorithm; (ii) problem-specific heuristics may be developed to yield high-quality solutions of large-size problems within a shorter time; and (iii) we may extend the proposed model by considering other practical issues, such as the spatial decision of lane reservation. References Bouillet, E., 2007. Path routing in mesh optical networks. John Wiley & Sons. Che, A., Wu, P., Chu, F., Zhou, M., 2015. Improved quantum-inspired evolutionary algorithm for large-size lane reservation. IEEE Transactions on Systems, Man, and Cybernetics: Systems 45 (12), 1535–1548. Dijkstra, E. W., 1959. A note on two problems in connexion with graphs. Numerische mathematik 1 (1), 269–271. Fang, Y., Chu, F., Mammar, S., Che, A., 2013. An optimal algorithm for automated truck freight transportation via lane reservation strategy. Transportation Research Part C: Emerging Technologies 26, 170–183. Goksal, F. P., Karaoglan, I., Altiparmak, F., 2013. A hybrid discrete particle swarm optimization for vehicle routing problem with simultaneous pickup and delivery. Computers & Industrial Engineering 65 (1), 39–53. Karp, R. M., 1972. Reducibility among combinatorial problems. In: Complexity of computer computations. Springer, pp. 85–103. Princeton, J., Cohen, S., 2011. Impact of a dedicated lane on the capacity and the level of service of an urban motorway. Procedia - Social and Behavioral Sciences 16 (16), 196–206. Prins, C., 2004. A simple and effective evolutionary algorithm for the vehicle routing problem. Computers & Operations Research 31 (12), 1985–2002. Shaabani, H., Kamalabadi, I. N., 2016. An efficient population-based simulated
18
annealing algorithm for the multi-product multi-retailer perishable inventory routing problem. Computers & Industrial Engineering 99, 189–201. Waxman, B. M., 1988. Routing of multipoint connections. IEEE journal on selected areas in communications 6 (9), 1617–1622. Wu, P., Che, A., Chu, F., Fang, Y., 2016. Exact and heuristic algorithms for rapid and station arrival-time guaranteed bus transportation via lane reservation. IEEE Transactions on Intelligent Transportation Systems, DOI 10.1109/TITS.2016.2631893. Wu, P., Che, A., Chu, F., Zhou, M., 2015. An improved exact -constraint and cut-and-solve combined method for biobjective robust lane reservation. IEEE Transactions on Intelligent Transportation Systems 16 (3), 1479–1492. Wu, Y., Chu, C., Chu, F., Wu, N., 2009. Heuristic for lane reservation problem in time constrained transportation. In: 2009 IEEE International Conference on Automation Science and Engineering. IEEE, pp. 543–548. Xiao, Y., Thulasiraman, K., Xue, G., J¨ uttner, A., 2005. The constrained shortest path problem: algorithmic approaches and an algebraic study with generalization. AKCE International Journal of Graphs and Combinatorics 2 (2), 63–86. Yen, J. Y., 1971. Finding the k shortest loopless paths in a network. Management Science 17 (11), 712–716. Zhou, Z., Chu, F., Che, A., Zhou, M., 2013. -constraint and fuzzy logic-based optimization of hazardous material transportation via lane reservation. IEEE Transactions on Intelligent Transportation Systems 14 (2), 847–857.
19
An automated truck freight transportation planning problem via lane reservation is studied.
Improved formulation is provided and several special cases of the problem are investigated.
An efficient two-phase exact algorithm based on problem properties is developed.
Computational results confirm the efficiency of the proposed model and algorithm.
S0360-8352(17)30172-9 http://dx.doi.org/10.1016/j.cie.2017.04.030 CAIE 4717
To appear in:
Computers & Industrial Engineering
Received Date: Revised Date: Accepted Date:
22 September 2016 25 February 2017 14 April 2017
Please cite this article as: Wu, P., Chu, F., Che, A., Fang, Y., An efficient two-phase exact algorithm for the automated truck freight transportation problem, Computers & Industrial Engineering (2017), doi: http://dx.doi.org/10.1016/ j.cie.2017.04.030
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
An efficient two-phase exact algorithm for the automated truck freight transportation problem Peng Wua,b , Feng Chub,c,∗, Ada Chea , Yunfei Fangd a
School of Management, Northwestern Polytechnical University, 710072 Xi’an, China b Laboratory IBISC, University of Evry-Val d’Essonne, 91020 Evry, France c Management Engineering Research Center, Xihua University, 610039 Chengdu, China d School of Economics & Management, Fuzhou University, 350116 Fuzhou, China
Abstract Fang et al. (2013) have developed an integer linear program and an exact algorithm for the automated truck transportation freight problem with lane reservation. However, due to its NP-hard nature, their proposed method becomes difficult to solve large-size problems within acceptable time. In this paper, we firstly present an improved integer linear program by adding valid inequalities and identify that its several special cases are classical combinatorial optimization problems. Based on analyzed properties, a new efficient two-phase exact algorithm is developed. Computational results on benchmark and new larger-size instances with up to 700 nodes and 55 tasks show that the new algorithm outperforms very favorably the state-of-the-art ones. Keywords: transportation, automated truck, lane reservation, integer programming, exact algorithm 1. Introduction Efficient shipments of cargos has attracted much attention and considerable freight transportation planning problems have been investigated extensively over the past decades (Prins, 2004; Goksal et al., 2013; Shaabani and Kamalabadi, 2016). However, increasing travel demand results in increasingly severe congestion, which causes many problems in transportation, such as low efficiency, unpredictable transport time, traffic accidents, and fuel ∗
Corresponding authors: [email protected] (F. Chu)
Preprint submitted to Computers & Industrial Engineering
May 13, 2017
waste. These problems increasingly prevent the freight transportation from being operated in an efficient, reliable and safe fashion (Fang et al., 2013). Introducing automated driving for trucks would be a promising solution to cope with such challenge, as automated trucks could provide remarkable advantages such as high safety and efficiency, and lower fuel consumption. Unlike manually driven trucks, automated ones must have the ability of detecting possible dangers and responding to them correctly and promptly. Dedicated truck lanes would be ideal in this sense. Since constructing new network dedicated to automated trucks may be infeasible due to the high costs and limited geographic space, converting existing general-purpose (GP) lanes in the existing network to dedicated ones is an effective alternative. But due to the exclusive use of reserved lanes by automated trucks, the available lanes in the network for GP vehicles are reduced, and negative impact, such as the increase in travel time of GP vehicles, will be generated on the adjacent lanes. It is necessary to well decide appropriate lanes to be reserved to achieve the safe and time-guaranteed automated truck transportation, while minimizing the negative traffic impact. Such an optimization problem is called the automated truck transportation problem with lane reservation (ATP) (Fang et al., 2013). We note that there have also been studies investigating lane reservation for other applications, such as large sport events, hazardous material transportation, bus transit (Wu et al., 2009; Zhou et al., 2013; Che et al., 2015; Wu et al., 2015, 2016). Fang et al. (2013) have formulated an integer linear program (ILP) and developed an exact cut-and-solve algorithm for the ATP. However, due to the NP-hardness of the ATP, their proposed methods become difficult to solve large-size problems within acceptable computational time. In this paper, we first provide an improved ILP by adding valid inequalities. Then, we identify that several special cases of the ATP are classical combinatorial optimization problems. Based on the analyzed properties, a new efficient two-phase exact algorithm is developed. Computational results on 120 benchmark and 210 new larger-size instances with up to 700 nodes and 55 tasks confirm the effectiveness of the proposed algorithm. The remainder of the paper is organized as follows. Section 2, recalls the problem description and provides the improved ILP. In Section 3, we derive several optimal properties of the ATP. Based on them, a new efficient exact algorithm is presented in Section 4. Section 5 reports the computational results. Section 6 concludes this study.
2
2. Problem description and formulation The ATP considered in this study has been addressed by Fang et al. (2013). For the sake of self-consistency, the problem description is first recalled as follows. The ATP can be defined on a transportation network that can be represented by a directed graph G(N, A) with a node set N and an arc set A. A node (resp. an arc) represents a road intersection (resp. a road segment). Given a set of automated truck transportation tasks to be accomplished and their corresponding origin-destination (OD) pair, the ATP consists of optimally selecting lanes to be reserved in the existing network and designing a reserved path for each task in order to ensure that it can be completed within its travel deadline safely. However, such lane reservation reduces the available lanes of GP vehicles such that the negative impact such as the increase of travel time on the adjacent lanes may be caused. The objective is to minimize the total negative impact caused by reserved lanes. As stated in Fang et al. (2013), some assumptions are made to facilitate the formulation of the ATP. First, the negative impact caused by a reserved lane is assumed as the increase of travel times on the remaining GP lanes. More details on the impact parameters can be found in Section 5. Second, at most one reserved lane is allowed on each road segment. Third, there is only one path for each task from its origin to destination in order to ensure the transport safety and the path only consists of reserved lanes. Fourth, there are at least two lanes on each road segment allowing one reserved lane. We first summarize the parameters and decision variables in Table 1.
3
Table 1: Notation and its explanation for the formulation
Notation Explanation G = {V, A} transportation network, where N and A denote the set of nodes and arcs, respectively i, j node index, i, j ∈ N k task index, k ∈ K O set of transportation tasks with |K| tasks D set of destination nodes, D ⊆ N ok origin node of task k ∈ K, ok ∈ O dk destination node of task k ∈ K, dk ∈ D Tk travel deadline of task k ∈ K Tij travel time on a reserved lane on arc (i, j) ∈ A Cij negative impact caused by a reserved lane on arc (i, j) ∈ A Zij equal to 1 if arc (i, j) is reserved, and 0 otherwise, (i, j) ∈ A Xkij equal to 1 the path of task k pass arc (i, j) which is reserved, and 0 otherwise, (i, j) ∈ A, k ∈ K 2.1. Problem formulation Before giving an improved model, we first recall the existing ILP for the ATP proposed by Fang et al. (2013), shown as follows. X IP : min Cij Zij (1) (i,j)∈A
X
s.t.
Xkok i = 1, for all k ∈ K
(2)
(ok ,i)∈A
X
Xkidk = 1, for all k ∈ K
(3)
(i,dk )∈A
X
Xkij =
j:(i,j)∈A
X
X
Xkji , for all j ∈ N \{ok , dk }, and all k ∈ K (4)
j:(j,i)∈A
Xkij Tij ≤ Tk , for all k ∈ K
(5)
(i,j)∈A
Xkij ≤ Zij , for all (i, j) ∈ A, and all k ∈ K Zij ∈ {0, 1}, for all (i, j) ∈ A Xkij ∈ {0, 1}, for all (i, j) ∈ A and all k ∈ K
4
(6) (7) (8)
Objective (1) is to minimize the total negative impact of all reserved lanes. Constraints (2)-(4) guarantee that there exists a feasible path for each OD pair. To be more specific, constraint (2) (resp. (3)) implies that there exists only one arc outgoing from (resp. coming into) origin node ok (resp. destination node dk ). Constraint (4) ensures the flow conservation for intermediate nodes between origin and destination for each task k ∈ K. Constraint (5) indicates that the total travel duration for task k from its origin to destination should not exceed its travel deadline. Constraint (6) ensures that task k can pass a reserved lane on arc (i, j) ∈ A only if this arc is reserved. Constraints (7) and (8) enforce the bounds of all decision variables. The benefit to be achieved for the ATP is the time-efficient automated truck transport service, i.e., each task should be completed within the given transportation deadline. Based on this goal, constraint (5) is formulated to specify the requirement of time-efficient transportation. For the ATP, the transport delay is not accepted, but trucks are allowed to arrive early since they achieve higher transport efficiency. We note that the model IP can be adapted to the case that does not allow trucks arrive too early but allows an acceptable delay by changing constraint (5) to a time window constraint. 2.2. An improved integer linear program for the ATP We improve the above formulation based on the following observations. Observation 1. For any task k ∈ K, there will be no arcs entering into (resp. outgoing from) its origin ok (resp. destination dk ) on its path. With Observation 1, we add the following valid inequalities into IP without excluding optimal solutions. Xkiok = 0, for all (i, ok ) ∈ A and all k ∈ K Xkdk i = 0, for all (dk , i) ∈ A and all k ∈ K
(9) (10)
Obviously, constraints (9) and (10) reduce the search space of the original problem since part of variables are prefixed. Observation 2. For any task k ∈ K, each node in the network will be passed at most once. Note that if a node in the network is passed more than once by a task (i.e., cycles exist on the transport path), this obviously generates larger negative 5
impact compared with the case without cycles. With Observation 2, we also add the following constraints into IP . X Xkij ≤ 1, for all j ∈ N \{ok , dk } and all k ∈ K (11) j:(i,j)∈A
X
Xkji ≤ 1, for all j ∈ N \{ok , dk } and all k ∈ K
(12)
j:(j,i)∈A
Constraints (11) and (12) are also valid inequalities, which tighten the search space of the original problem. With the newly obtained constraints, we derive the following improved IP model. X IP 0 : min Cij Zij (i,j)∈A
s.t. s.t. Constraints (2)-(12) P + Remark 1. The improved formulation employs k (|A− k |+|Ak |)+2|K|(|N |− 2) more constraints as compared with Fang et al. (2013)’s formulation, where + |A− k |(resp. |Ak | denotes the number of arcs entering into origin ok (resp. outgoing from destination dk ). P + Remark 2. The prefixing of k (|A− k | + |Ak |) variables and relatively more constraints help reduce search space. Computational results in Section 5.1 show that IP’ can save 19.11% average time compared with IP. Fang et al. (2013) stated that the ATP is NP-hard based on their observation but the proof was not given. The detailed proof is given below. Theorem 1. The ATP is NP-hard. Proof. The proof is based on the fact that constrained shortest path problem (CSPP) reduces to the special case of the ATP with only one task. Let us consider the following instance of CSPP. Given a graph G0 = (N 0 , A0 ) associated with two weights cij > 0 (say, cost) and dij > 0 (say, delay), (i, j) ∈ A0 , two distinguished nodes s and t, s, t ∈ N 0 and a positive value T . The CSPP consists of finding a minimum cost s-t path with its total delay being equal to or less than T . We then show how to transform the above CSPP instance into an instance of the special case of the ATP with only one task. Denote the only task as 6
task 1. Let us map (N 0 , A0 ) to (N, A) (i.e., N 0 and A0 are mapped to N and A, respectively), s to o1 and t to d1 . Cij , Tij , and T1 are equal to cij , dij , and T , respectively. Through such linear transformation, the CSPP is reduced into the special case of the ATP with only one task. As the CSPP is NP-hard even for acyclic networks (Xiao et al., 2005), the ATP with one task is consequently NP-hard. Certainly, the ATP in general case is NP-hard. 3. Property analysis for the ATP In this section, we first investigate several special cases for the ATP. Note that these special cases correspond to classical combinatorial optimization problems and can be tackled using existing techniques. The potential benefits are that if an instance is recognized as one special case of them, then it can be efficiently solved accordingly. Then, the ATP in the general case is analyzed. 3.1. Special cases of the ATP Case 1: The ATP with only one task and large task travel deadline. When the travel deadline of the task is large enough, then the travel deadline constraint can be relaxed. Obviously, the special ATP in Case 1 is equivalent to find a reserved path with minimum impact. Then, the following proposition is straightforward. Proposition 1. The special ATP in Case 1 is equivalent to a shortest path problem. Remark 3. The special ATP in Case 1 is polynomial solvable as the shortest path problem that can be efficiently solved by the Dijkstra shortest path algorithm (Dijkstra, 1959) whose time complexity is O(|N |2 ). Case 2: The single-source ATP with large task travel deadline. For this case, we have the following proposition. Proposition 2. The special ATP in Case 2 is equivalent to a directed Steiner tree problem (DSTP).
7
Proof. Let us consider an instance of DSTP as follows. Given a directed graph G = (V, E) with positive weights on the edges, a set of terminal vertices V 0 ⊆ V , a root node vertex r. i, j ∈ V , edge (i, j) ∈ E and its weight is denoted by wij . The instance of DSTP consists of finding a minimum weight out-branching tree T rooted at r, such that the vertices in V 0 are included in T . We now show how to transform the DSTP instance into an instance of the special ATP in Case 2. Let us map graph G = (V, E) to graph G = (N, A) (i.e., V and E are mapped to N and A, respectively), V 0 to D and r to the only origin node. The weight Cij is equal to wij . Then, designing the task paths with minimal impact of lane reservation is equivalent to finding a minimum weight out-branching tree T . With such linear transformation, the DSTP is reduced into the special Case 2 of ATP. Remark 4. The special ATP in Case 2 is NP-hard, as the DSTP is known to be NP-hard (Karp, 1972). Case 3: The ATP with only one task. By the proof of Theorem 1, we have the following proposition. Proposition 3. The special ATP in Case 3 is equivalent to a constrained shortest path problem. Remark 5. The special ATP in Case 3 is NP-hard. 3.2. The ATP in general case Generally, an ATP contains multiple tasks and different tasks are allowed to share reserved lanes. Since the path design of any task may be influenced by other tasks’ paths due to the requirement of minimizing negative impact of reserved lanes. Consequently, the path of any task may be not an optimal path determined by solving its corresponding constrained shortest path problem. That is to say, solving an ATP with |K|(|K| > 1) tasks may be not equivalent to independently solving |K| constrained shortest path problems. For the ATP in general case, the following proposition is straightforward. Proposition 4. Solving an ATP with multiple tasks is equivalent to finding an optimal loopless path respecting travel deadline constraint for each task to form the best path combination such that the total negative impact of reserved lanes is minimized. 8
As previously analyzed, we know that the ATP with only one task is NPhard even for acyclic networks. This means that the ATP with multiple tasks in the network with cycles is even harder to handle. In the following section, an exact method to efficiently solve the ATP in general case is developed. 4. Two-phase exact algorithm for the ATP For an ATP in general case, Fang et al. (2013) proposed a cut-and-solve algorithm, which can solve problem instances with up to 150 nodes in the network and 30 tasks within 18000 CPU seconds. In this paper, a new efficient two-phase exact algorithm is developed to efficiently solve the largersize ATP. The algorithm is composed of two major phases. In the first phase, all feasible paths respecting the travel deadline constraint are enumerated for each task k ∈ K. An optimal lane reservation scheme and task paths are then determined in the second phase. We detail our new optimal algorithm in what follows. 4.1. Phase 1: Task path enumeration As indicated by Proposition 3, the ATP with one task is a constrained shortest path problem that consists of finding a loopless reserved path respecting the travel deadline constraint with minimal negative impact. Thus, for any task k ∈ K, let Pk denote the set of all loopless paths connecting its origin and destination and respecting its travel deadline (i.e., the total travel duration is equal to or less than Tk ). Remark 6. The optimal path of task k ∈ K of the ATP must be in set Pk . It is not hard to find that the problem determining the set Pk for each task k ∈ K is equivalent to finding all loopless paths with their travel duration being equal to or less than Tk in the direct graph G(N, A). We note that such problem can be efficiently solved using the well-known Yen’s K-shortest loopless path algorithm (Yen, 1971) with its time complexity O(K|N |(|A| + |N |log|N |) (Bouillet, 2007), which belongs to a kind of deviation algorithms ranking the first K loopless paths for given pair of nodes. To more efficiently obtain the set Pk , ∀k ∈ K, the search space for each k ∈ K is reduced before using Yen’s K-shortest loopless path algorithm. For k ∈ K, a ∈ A, we define the possibly passed arc set Ak as follows. Ak = {(i, j)|ϕ(ok , i) + Tij + ϕ(j, dk ) ≤ Tk }, k ∈ K 9
(13)
where ϕ(ok , i)(resp. ϕ(j, dk ) denotes the shortest path from ok to i(resp. j to dk ) when all the arcs in the network are reserved. Note that arcs belonging to A\Ak would not be passed by task k, otherwise the travel deadline constraint will be violated. In other words, set Ak includes all arcs in the network that may be passed by task k ∈ K. Since set Ak is a subset of A, obviously the resolution of the path enumeration for task k will be accelerated. Remark 7. From Proposition 4, we can observe that the path selected for task k ∈ K in the optimal solution of the ATP may not be the travel deadline constrained path with minimal negative impact. As the above remark, after obtaining the set Pk for each k ∈ K, the path for any task cannot be simply determined to be the one in the set Pk with the minimal negative impact. According to Proposition 4, the ATP is to find the best task path combination so as to minimize the total impact. 4.2. Phase 2: Lane reservation and task path determination The candidate path sets Pk , ∀k ∈ K are determined in the above phase. In the second phase, we propose an integer programming method to determine the optimal lane reservation scheme and task path from Pk for each task k ∈ K. To formulate the model of determining optimal lane reservation and k all task paths, we need to additionally define parameter δpij , if path p ∈ Pk k passes arc (i, j) ∈ Ak , δpij = 1, and 0 otherwise, and a new variable Ypk below. Ypk : = 1 if the path p ∈ Pk is selected; and 0 otherwise, ∀k ∈ K. Then, the new ILP for determining optimal lane reservation and task path is given as follows. X IP 00 : min Cij Zij (i,j)∈A
X
s.t.
k δpij Ypk
≤ Zij , for all k ∈ K, and all (i, j) ∈ A
(14)
p∈Pk
X
k Ypk ≤ Zij , for all k ∈ K, and all (i, j) ∈ A δpij
(15)
p∈Pk
Ypk , Zij ∈ {0, 1}, ∀p ∈ Pk , for all k ∈ K, and all (i, j) ∈ A
(16)
where constraint (14) indicates that only one path is selected from the candidate path set Pk . Constraint (15) ensures that the path of task k, ∀k ∈ K 10
involving arc (i, j), ∀(i, j) ∈ A only if this arc is reserved. Constraint (16) gives the ranges of decision variables. Note that IP 00 is an integer linear program which can be tackled by the optimization software such as CPLEX. Algorithm 1 Two-phase exact algorithm for the ATP 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11:
initialize Pk = ∅ for all k ∈ K, k=0; while k ≤ |K|, do: let l = 1; for task k, define set Ak based on formula (13); for OD pair (ok , dk ), compute the lth shortest path using Yen’s K-shortest loopless path algorithm, and record the travel duration dkl and its path pkl ; if dkl ≤ Tk , Pk = Pk ∪ {pkl }, l = l + 1 and goto step 5; otherwise k = k + 1 and goto step 2; end while construct model IP 00 using sets Pk , ∀k ∈ K obtained above; solve IP 00 exactly using CPLEX IP solver; output the lane reservation scheme and the corresponding paths for tasks.
4.3. Overall algorithm for the ATP The overall algorithm for the ATP can be outlined in Algorithm 1, in which steps 1-8 (i.e., Phase 1) aim to enumerate all feasible paths for each task and steps 9-11 are to determine the final lane reservation scheme and task paths (i.e., Phase 2). 5. Computational experiments In this section, we conduct numerical computational experiments to show the performance of the proposed algorithm. Our algorithm is coded in C++ language and combined with Yen’s K-shortest loopless path algorithm (Yen, 1971) and CPLEX (12.6) IP solver with default settings. All the experiments are done on a PC with 2.5 GHz and 2.95 GB RAM with windows 7 system. The performance of the proposed algorithm is evaluated on eighty-one groups of instances with five instances each group, including 120 benchmark instances (Fang et al., 2013) and 210 new larger-size instances. These new instances are randomly generated based on the way in Wu et al. (2009) and Fang et al. (2013). Waxman’s network model Waxman (1988) is used to 11
generate the network G(N, A). D = 2|A|/|N | is called average node degree. To be more specific, the nodes of G(N, A) are randomly distributed in a square area [0, 100] × [0, 100], the existence of arc (i, j) between nodes i and j is dependent by a probability function a exp (−Lij /bLmax ), where Lij and L are the Euclidean distance between nodes i and j and the maximum Euclidean distance between any pair of nodes, respectively, and 0 ≤ a ≤ 1 and 0 ≤ b ≤ 1. The origin and destination nodes ok and dk , k ∈ K are randomly generated from set N . Tij is computed by Lij /Vij , where Vij denotes the average travel speed on a reserved lane of arc (i, j) and is set as 60 km/h. Let Tij0 denote the travel time on arc (i, j) without reserved lanes, which is approximately computed by the widely used BPR function, defined as Tij0 = Tij (1 + α(fij /capij )β )), where fij and capij are the traffic flow per unit time within the considered time horizon and capacity of arc (i, j), respectively. According to Wu et al. (2015), fij /capij is randomly generated in [0.5, 1.2] and α = 0.15 and β = 4. The impact of a reserved lane on arc (i, j) is defined as Cij = Tij0 /(Mij −1), where Mij is the total number of lanes on arc (i, j), as is the case in Wu et al. (2009) and Fang et al. (2013). Therefore, the negative impact due to a reserved lane is proportional to the travel time without reserved lanes and inversely proportional to the number of lanes. If no lane is reserved, then Cij =0. As indicated in Princeton and Cohen (2011), an actually statistical result showed that the travel time of the non-reserved lanes increased about 53% after one of the three lanes was reserved in A1 highway in Paris, which is close to the theoretical result (50%) obtained by the above formula. Tk is defined as Lk + λk (L0k − Lk ), where Lk (resp. L0k ) is the shortest travel time from origin ok to destination dk when all arcs in the network are reserved (resp. no lanes in the network are reserved) and λk is a given parameter, which is randomly generated in [0, 1]. For brevity, let Tcp , Tcs and Ttp denote the average computational time (CPU seconds) spent by CPLEX IP solver solving IP 0 (i.e., solving the ATP with a direct use of CPLEX), the CS algorithm (Fang et al., 2013) and by our two-phase algorithm for five instances of each group, respectively. Besides, let Ttp-f (resp. Ttp-s ) denote the average CPU time (seconds) spent by the first (resp. second) phase of our proposed algorithm. Note that the computational times of both methods are limited to 18000s (i.e., five hours), as is the case in Fang et al. (2013).
12
5.1. Model comparison To demonstrate that the proposed improved model (see Section 2.2) is more efficient, we compare it with the model proposed by Fang et al. (2013) (Fang’s model in short) by solving a number of instances. The two models for these instances are both solved by CPLEX. The comparison results are summarized in Table 2. Saved time rate means the reduced time rate by the improved model compared with the one proposed by Fang et al. (2013). We can observe from Table 2 that the computational time of the improved model is less than that of Fang’s model and the former saves an average 19.11% time compared with the latter. This indicates that the improved model IP 0 is more efficient than the model proposed by Fang et al. (2013). Table 2: Performance of the improved model Group 1 2 3 4 5 6 7 8 Average
|N | 60 60 70 70 80 80 90 90
|K| 25 30 25 30 25 30 25 30
D 7 7 7 7 7 7 7 7
Time of Fang’s model (s) 9.95 26.73 40.99 93.11 46.62 124.76 171.08 317.12 103.80
Tcp 6.75 22.89 36.25 64.78 36.19 102.64 99.24 302.96 83.96
Saved time rate(%) 32.09 14.37 11.56 30.43 22.36 17.73 41.99 4.47 19.11
5.2. Results for benchmark instances in Fang et al. (2013) To evaluate the performance of the proposed algorithm, we first compare it with benchmark instances with the state-of-the-art algorithm (i.e., the CS algorithm proposed by Fang et al. (2013)) in terms of computational time in obtaining optimal solutions. The results are reported in Tables 3 and 4. Table 3 reports the results for instances with fixed number of nodes |N | = 100 and average node degree D ranging from five to twelve. It can be observed from Table 3 that the CPU time of the CS algorithm Tcs varies from 1.59s to 8900.67s with its average value being 873.18s, while that of our algorithm Ttp varies from 0.62s to 1080.95s with its average value being 85.36s. Ttp is less than Tcs over all groups 9-23 except the small-size groups 9, 10 and 12. On average, Tcs is eleven times more than Ttp . This indicates that our algorithm is much more efficient than the CS algorithm in terms of computational time. Moreover, Tcs increases sharply with |K| for a given average node degree, 13
Table 3: Comparison results for instances with |N |=100 and various node degrees Group 9 10 11 12 13
|N | 100 100 100 100 100
|K| 10 15 20 25 30
D 5 5 5 5 5
Tcs 1.59 1.56 41.82 37.21 63.45
Ttp-f 3.42 1.52 8.68 57.98 3.80
Ttp-s 0.39 0.35 0.95 7.05 1.09
Ttp 3.80 1.87 9.64 65.03 4.89
Tcs /Ttp 0.42 0.84 4.34 0.57 12.98
14 15 16 17 18
100 100 100 100 100
10 15 20 25 30
7 7 7 7 7
6.41 62.32 315.98 1193.88 1288.72
0.52 2.46 2.87 1.75 1.75
0.09 0.26 0.32 0.24 0.30
0.62 2.72 3.19 1.98 2.05
10.38 22.88 99.10 601.65 627.73
19 20 21 22 23 Average
100 100 100 100 100
10 15 20 25 30
12 12 12 12 12
18.67 78.30 496.89 1590.22 8900.67 939.85
2.45 5.79 67.25 10.61 197.38 24.55
0.22 0.36 16.23 0.77 883.58 60.81
2.67 6.14 83.48 11.38 1080.95 85.36
7.00 12.75 5.95 139.72 8.23 11.01
especially for larger D = 7 and 12, whereas Ttp varies slightly and it does not necessarily increase with |K|. For example, Ttp for groups 8 and 9 decreases from 3.19s to 1.98s when |K| increases from 20 to 25. On the other hand, we observe that Ttp-s are positively correlated with Ttp-f over groups 1-15 (i.e., the larger Ttp-f is, the larger Ttp-s is). For example, Ttp-f ’s are 8.68s and 57.98s for groups 3 and 4, respectively, and Ttp-s ’s are 0.95s and 7.05s, respectively. The main reason is that a larger Ttp-f implies that more task path candidates exist for the tasks, which means more integer variables Ypk in IP 00 in step 5 of our algorithm such that it is more difficult to be solved. Besides, it can be seen that generally Ttp-f is larger than Ttp-s , which shows the first phase consumes more time. An exception is group 23 (i.e., the largest-size group) where Ttp-s is larger than Ttp-f . The reason is that there exist many task path candidates and it makes the model IP 00 formed in second phase much difficult to be solved. Table 4 presents the results when |N | is increased from 110 to 150 and |K| ranges from 10 to 30 for a given average node degree. In Table 4, we can see that Tcs varies from 6.27s to 1878.07s with its average value being 788.46, whereas Ttp ranges from 5.70s to 108.31s with it average value being 21.01s. Ttp is less than Tcs over groups 24-32, except the smallest-size group 14
Table 4: Comparison results for instances with fixed node degree and |N |= 110-150 Group 24 25 26 27 28 29 30 31 32 Average
|N | 110 110 120 120 130 130 140 140 150
|K| 10 15 15 20 20 25 25 30 30
D 7 7 7 7 7 7 7 7 7
Tcs 6.27 18.02 103.29 121.68 299.71 1406.91 1575.25 1686.95 1878.07 788.46
Ttp-f 6.07 7.93 16.20 4.56 7.68 9.36 5.19 17.05 100.21 19.36
Ttp-s 0.58 0.85 1.70 0.55 0.59 0.61 0.51 1.31 8.09 1.64
Ttp 6.65 8.78 17.90 5.11 8.28 9.97 5.70 18.36 108.31 21.01
Tcs /Ttp 0.94 2.05 5.77 23.81 36.21 141.09 276.35 91.89 17.34 37.54
24. The CS algorithm spends more than 37 times average time of that by our algorithm. This further shows the efficiency of our algorithm as compared with the CS algorithm proposed by Fang et al. (2013). 5.3. Results for new larger-size instances In order to further evaluate the performance of the proposed method for solving larger-size problems, 42 new larger-size problem groups are tested. Computational results are reported in Tables 5-7. Table 5: Computational results for instances with |N |= 160-200 Group 33 34 35 36 37 38 39 40 41 42 Avg.
|N | 160 160 170 170 180 180 190 190 200 200
|K| 20 30 30 35 35 40 40 45 45 50
Tcp 889.01 4312.34 13201.26 16334.21 >14273.68
Tcs 368.72 2432.11 6848.21 11329.37 17582.12 >12856.05
Ttp-f 8.79 11.04 2.75 2.89 2.65 2.58 7.13 8.21 5.13 5.82 5.70
Ttp-s 1.02 0.69 0.22 0.24 0.32 0.37 0.86 0.95 0.41 0.45 0.55
Ttp 9.81 11.73 2.97 3.13 2.97 2.95 7.99 9.16 5.54 6.27 6.25
Table 5 presents the results of large-size instances with |N | increasing from 160 to 200, |K| varying from 20 to 50 and D=7. We can observe in Table 5 that the CPU time consumed by our algorithm is far less than those by CPLEX and the CS algorithm over all groups 33-42. As the problem size increases, Tcp and Tcs increase exponentially, whereas Ttp varies slightly. It 15
is worthwhile to note that CPLEX and the CS algorithm can only exactly solve the four (resp. five) of ten groups within 18000s, whereas our algorithm can exactly solves all the groups with an average time 6.25s. Besides, we can observe that Ttp-f is larger than Ttp-s over all the groups 33-42. Table 6: Computational results for instances with |N |= 300-700 Group 43 44 45 46 47 48 49 50 51 Avg. 52
|N | 300 300 400 400 500 500 600 600 700
|K| 30 40 40 45 45 50 50 55 55
D 5 5 5 5 5 5 5 5 5
700
60
5
53 54 55 56 57 58 Avg. 59
300 300 400 400 500 500
30 40 40 45 45 50
10 10 10 10 10 10
600
50
10
Tcp -
Tcs -
Ttp-f 24.02 26.51 32.22 33.73 41.73 65.12 295.27 741.29 1753.30 334.80 -
Ttp-s 1.75 2.27 1.99 1.93 17.27 23.17 454.89 627.30 6699.99 870.06 -
Ttp 25.77 28.78 34.21 35.66 59.00 88.29 750.16 1368.59 8453.29 1204.86 -
-
-
69.11 248.37 210.23 170.02 869.55 765.37 388.78 -
5.99 20.75 16.25 7.57 99.91 85.12 39.27 -
75.10 269.12 226.48 177.59 969.46 850.49 428.04 -
Table 6 reports the computational results of larger-size instances with |N | increasing from 300 to 700, |K| varying from 30 to 60 under two scenarios D=5 and 10. It can be found that CPLEX and the CS algorithm by Fang et al. (2013) cannot generate optimal solutions for any group within 18000s (i.e., they lose their power for these large-size instances totally), whereas the proposed algorithm is able to exactly solve the instances with up to 700 nodes and 55 tasks with average node degree D = 5 and 500 nodes and 50 tasks with average node degree D = 10, respectively. Due to the NP-hardness of the problem, we find that the proposed algorithm rapidly increases with the problem size for each given average node degree. For example, Ttp for group 43 is 25.77s, while for group 51 is 8453.29s. Moreover, it is not hard to find that the increase of Ttp is mainly caused by the increase of Ttp-s . We also observe that our algorithm consumes more time to solve instances with 16
Table 7: Comparison for sensitivity analysis of Cij = rij Tij0 /(Mij − 1) Group 60 61 62 63 64 Avg. 65 66 67 68 69 Avg. 70 71 72 73 74 Avg.
rij [0.5, 1) [0.5, 1) [0.5, 1) [0.5, 1) [0.5, 1)
|N | 200 300 400 500 600
|K| 20 20 20 20 20
1.0 1.0 1.0 1.0 1.0
200 300 400 500 600
20 20 20 20 20
200 300 400 500 600
20 20 20 20 20
(1, (1, (1, (1, (1,
1.5] 1.5] 1.5] 1.5] 1.5]
Ttp-f 3.34 6.11 13.12 17.35 69.99 21.98 2.59 3.88 17.15 30.24 67.44 24.26 3.56 5.22 14.98 17.11 80.01 24.18
Ttp-s 0.27 0.45 0.99 1.93 3.99 1.53 0.32 0.53 1.30 2.41 5.74 2.06 0.32 0.46 0.92 1.31 4.65 1.53
Ttp 3.61 6.56 14.11 19.28 73.98 23.51 2.91 4.41 18.45 32.65 73.18 26.32 3.88 5.68 15.90 18.42 84.66 25.71
a larger average node degree when |N | and |K| are given. For example, Ttp for group 48 is 88.29s, while for group 58 is 850.49s. The possible reason is that a larger average node degree may result in more task path candidates for each task, which requires more computational effort for both phases in the proposed algorithm. We note that the proposed algorithm loses power for solving groups 52 and 59 due to the lack of memory in the first phase. We have also conducted experiments to evaluate the sensitivity of different impact parameters on the performance of the proposed algorithm. Table 7 reports the computational results. Note that the intervals [0.5, 1) and (1, 1.5] are used to generate smaller and larger impact parameters, respectively. It can be seen that the ranges of Ttp under the three scenarios are 3.61-73.98s, 2.91-73.18s, and 3.88-84.66s, respectively. Moreover, Tcs ’s under the three scenarios are 23.51s, 26.32s, and 25.71s, respectively. These results indicate that our algorithm is insensitive to the changes of Cij . 6. Conclusion In this paper, we have revisited the automated truck transportation problem with lane reservation proposed by Fang et al. (2013). For the problem, we first propose valid inequalities for the integer linear program proposed 17
by Fang et al. (2013). Computational comparison results indicate that these valid inequalities are effective in saving computational time. Furthermore, we have investigated several special cases of the considered problem, which can be identified to be classical combinatorial optimization problems. To efficiently solve the problem, we devise a new efficient two-phase exact algorithm based on analyzed properties. Computational results demonstrate that the proposed algorithm significantly outperforms the state-of-the-art algorithm and it can solve larger-size instances with up to 700 nodes and 55 tasks. Future research directions may include the following aspects: (i) one important direction is to develop more efficient techniques for the resolution of IP 00 to further accelerate the proposed algorithm; (ii) problem-specific heuristics may be developed to yield high-quality solutions of large-size problems within a shorter time; and (iii) we may extend the proposed model by considering other practical issues, such as the spatial decision of lane reservation. References Bouillet, E., 2007. Path routing in mesh optical networks. John Wiley & Sons. Che, A., Wu, P., Chu, F., Zhou, M., 2015. Improved quantum-inspired evolutionary algorithm for large-size lane reservation. IEEE Transactions on Systems, Man, and Cybernetics: Systems 45 (12), 1535–1548. Dijkstra, E. W., 1959. A note on two problems in connexion with graphs. Numerische mathematik 1 (1), 269–271. Fang, Y., Chu, F., Mammar, S., Che, A., 2013. An optimal algorithm for automated truck freight transportation via lane reservation strategy. Transportation Research Part C: Emerging Technologies 26, 170–183. Goksal, F. P., Karaoglan, I., Altiparmak, F., 2013. A hybrid discrete particle swarm optimization for vehicle routing problem with simultaneous pickup and delivery. Computers & Industrial Engineering 65 (1), 39–53. Karp, R. M., 1972. Reducibility among combinatorial problems. In: Complexity of computer computations. Springer, pp. 85–103. Princeton, J., Cohen, S., 2011. Impact of a dedicated lane on the capacity and the level of service of an urban motorway. Procedia - Social and Behavioral Sciences 16 (16), 196–206. Prins, C., 2004. A simple and effective evolutionary algorithm for the vehicle routing problem. Computers & Operations Research 31 (12), 1985–2002. Shaabani, H., Kamalabadi, I. N., 2016. An efficient population-based simulated
18
annealing algorithm for the multi-product multi-retailer perishable inventory routing problem. Computers & Industrial Engineering 99, 189–201. Waxman, B. M., 1988. Routing of multipoint connections. IEEE journal on selected areas in communications 6 (9), 1617–1622. Wu, P., Che, A., Chu, F., Fang, Y., 2016. Exact and heuristic algorithms for rapid and station arrival-time guaranteed bus transportation via lane reservation. IEEE Transactions on Intelligent Transportation Systems, DOI 10.1109/TITS.2016.2631893. Wu, P., Che, A., Chu, F., Zhou, M., 2015. An improved exact -constraint and cut-and-solve combined method for biobjective robust lane reservation. IEEE Transactions on Intelligent Transportation Systems 16 (3), 1479–1492. Wu, Y., Chu, C., Chu, F., Wu, N., 2009. Heuristic for lane reservation problem in time constrained transportation. In: 2009 IEEE International Conference on Automation Science and Engineering. IEEE, pp. 543–548. Xiao, Y., Thulasiraman, K., Xue, G., J¨ uttner, A., 2005. The constrained shortest path problem: algorithmic approaches and an algebraic study with generalization. AKCE International Journal of Graphs and Combinatorics 2 (2), 63–86. Yen, J. Y., 1971. Finding the k shortest loopless paths in a network. Management Science 17 (11), 712–716. Zhou, Z., Chu, F., Che, A., Zhou, M., 2013. -constraint and fuzzy logic-based optimization of hazardous material transportation via lane reservation. IEEE Transactions on Intelligent Transportation Systems 14 (2), 847–857.
19
An automated truck freight transportation planning problem via lane reservation is studied.
Improved formulation is provided and several special cases of the problem are investigated.
An efficient two-phase exact algorithm based on problem properties is developed.
Computational results confirm the efficiency of the proposed model and algorithm.