2014 IEEE 17th International Conference on Intelligent Transportation Systems (ITSC) October 8-11, 2014. Qingdao, China
A bus lane reservation problem in urban bus transit network* Peng Wu, Feng Chu, Senior Member, IEEE, Ada Che, Member, IEEE, Qin Shi
Abstract—This work studies a new variant of lane reservation problem called a bus lane reservation problem. It aims to optimally select some road segments to be reserved for buses in a predetermined bus transit network such that bus travelers can be rapidly transited. Two realistic assumptions are made in this work: 1) the total travel time on each bus route should be completed within a given deadline; 2) a lane on a road segment can be reserved for buses only when the bus volume on it exceeds a given level. The former guarantees bus travel time in order to improve the service level of a bus transit system. The second ensures minimum bus utilization on a reserved lane. However, private vehicles are not allowed to pass the bus exclusive lanes. Such a bus lane reservation strategy causes negative traffic impact on private vehicles. The objective of the problem is to minimize the negative traffic impact of reserved lanes. For the problem, we first construct an integer linear programming model and demonstrate its complexity to be NP-hard. Then, we propose a cut-and-solve algorithm to obtain its optimal solution. The computational results on randomly generated instances show that the proposed method is more efficient than the optimization software CPLEX 12.4.
I. INTRODUCTION Traffic congestion is one of the major challenges all over the world. This is mainly caused by the ongoing and rapid growth of travel demand, whereas the capacity of transport networks is limited. Such imbalance in demand and supply causes adverse impacts on the urban transportation systems, such as long delay, high fuel waste, and heavy pollution. The most direct means of alleviating urban traffic congestion is to build new roads to increase the capacity of transport networks, but it is unfortunately restricted by the geographic space, high construction costs, long duration and environmental impact. The development of public transport is an important strategy * This work was partially supported by the Cai Yuanpei Program between the French Ministries of Foreign and European Affairs and the Higher Education and Research and the Chinese Ministry of Education under Grant No. 27927VE; by the program of 100 Foreign Experts in Anhui Province and the program of Chair professor of Huangshan Scholars at Hefei University of Technology; by the Humanities and Social Science Foundation of the Chinese Ministry of Education under Grant 12XJCZH004; by the Humanities, Social Sciences and Management Innovation Foundation of Northwestern Polytechnical University (RW201301); and by the National Natural Science Foundation of China under Grant 71431003. P. Wu is with the School of Management, Northwestern Polytechnical University, 710072 Xi’an, China, and also with the Laboratory IBISC, University of Evry-Val d’Essonne, 91020 Evry, France (e-mail: wupeng888
[email protected]). F. Chu is with the Laboratory IBISC, University of Evry-Val d’Essonne, 91020 Evry, France and with the school of Transportation Engineering, Hefei University of Technology, 230009 Anhui, China (e-mail: feng.chu @ufrst. univ- evry.fr). A. Che is with the School of Management, Northwestern Polytechnical University, 710072 Xi’an, China (e-mail:
[email protected]). Q. Shi is with the school of Transportation Engineering, Hefei University of Technology, 230009 Anhui, China(
[email protected]). 978-1-4799-6078-1/14/$31.00 ©2014 IEEE
to reduce traffic congestion for governments in many cities. Public transport will be a practical solution if it is efficient, effective, reliable and comfortable. Bus transit as one of the oldest public transport modes has great advantages in the high flexibility and low fare [1]. However, the bus transit is less attractive to passengers, which is mainly due to its unreliable service level. Time-efficient transit is one of the most important factors for evaluating the service level of a bus transit system. A lane reservation strategy converting some general-purpose lanes (GPLs) from an existing bus transit network into bus exclusive lanes has been widely applied to reduce buses’ travel time and improve the bus service level. Bus exclusive lanes reflect the priority of buses, which improves the travel speed of buses, and reduce their travel time. Choi and Choi [2] concluded that bus travel time was significantly reduced with a lane reservation strategy. Wei and Chong [3] found that the average speed of buses in Kunming, China increased up to 58%, from 9.6 km/hour to 15.2 km/hour with a two years’ bus exclusive lane operation. Shalaby [4] reported that bus exclusive lanes are useful in improving bus transit performance in an urban arterial road segment in the downtown of Toronto, Canada. However, private vehicles are not allowed to pass the reserved lanes so that the adjacent GPLs may more congested and increase the travel time on them. Princeton and Cohen [5] concluded that the average travel time on GPLs was increased up to 26% after a lane of A1 motorway in Paris was reserved. Hence, the reservation of bus exclusive lanes in a bus transit network must be carefully considered. A range of studies focuses on bus lane reservation problem in the literature. They could be classified into two categories. Studies in the first one focus on one or several road segment by evaluation methods. Black [6] proposed an evaluation model for bus exclusive lane implementation on urban arterial road segments. Seo et al. [7] established some guidelines to bus exclusive lane selection in Seoul. They concluded that bus exclusive lanes would be useful only when the condition of certain total traffic volume and bus volume levels is met. Gan et al. [8] presented an evaluation and decision model for bus exclusive lane selection on arterial road segments. Eichler and Daganzo [9] proposed a concept of intermittent bus exclusive lane on which private vehicles are allowed to pass when no buses are moving on it, and adopted the kinematic wave theory to study the bus exclusive lane selection on a long arterial road segment. However, the studies above only investigated bus lane reservation on a limited number of road segments and did not consider the optimal bus exclusive lane selection from the perspective of a bus transit network. Studies in the second category consider optimal bus lane reservation in an existing or predetermined transport network by optimization methods. Mesbah et al. [10] presented a
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framework to obtain an optimal lane reservation solution for buses in the bus transit network and considered the bus lane reservation problem as a Stackelberg leader-follower problem in which transport system managers act as the leader and the users act the followers. The problem was formulated as a bi-level programming model. The aim of the upper level is to minimize the travel time by both buses and private vehicles in a urban transportation network, and the low level is a constant modal split/demand assignment model that captures the reaction of drivers. Mesbah et al. [11] extended their model in [10] by integrating several other objectives, and proposed a decomposition method to find optimal solutions. Later, Mesbah et al. [12] applied a genetic algorithm (GA) to obtain near-optimal solutions for the problem in [11]. Li and Ju [13] studied the travelers’ reactions to the reserved bus lanes from mode selection, paths choice, and departure choice. They formulated a variational inequality model and designed a heuristic algorithm for it. The results show that when transport demands increase up to 3000-4000 persons/hour, bus passengers will benefit from a bus lane, while private vehicles sustain more delay and a mode shift from private vehicles to buses reduces the delay. Recently, Khoo et al. [1] proposed a bi-objective bus lane reservation and scheduling problem with the aim of simultaneously minimizing the travel time of buses and private vehicles, and applied NSGA II to find Pareto optimal solutions. Its objective values were calculated by a simulation tool called Paramics. These above studies are very interesting, but some limitations below exist: 1) all the proposed algorithms were evaluated by only one study case; 2) the size of study cases are limited in relatively small networks; 3) the proposed non-linear programming models are difficult to solve due to their nonlinearity nature for large-size instances; 4) negative impact of bus exclusive lanes on normal traffic have not been considered yet in these works. Different from the existing studies, this work studies a bus lane reservation problem from the perspective of minimizing negative traffic impact of reserved lanes. To the best of our knowledge, there are few studies that consider minimizing the total traffic reserved lanes’ impact in the literature. Wu et al. [14] firstly formulated an integer linear programming (ILP) model for a lane reservation problem (LRP). The problem aims to complete some special time-guaranteed transportation tasks via optimally reserving lanes during large sport events. Its objective is to minimize the negative impact of reserved lanes. They proposed a heuristic algorithm to obtain its near-optimal solutions. Later, Fang et al. [16]-[18] studied three variants of LRP that were all proved to be NP-hard and proposed exact cut-and-solve algorithms. Recently, Zhou et al. [19] addressed a hazardous material transportation via a lane reservation strategy, formulated a bi-objective integer linear programming model, and proposed an -constraint and fuzzy logic combined method. In this work, we study a new variant of LRP, called bus lane reservation problem (BLRP). It intends to optimally choose lanes to be reserved for buses in a predetermined bus transit network for time-guaranteed bus transit. For the considered problem, we suppose that the total bus route travel time from its origin to terminal must be completed within a given deadline. This aims to improve the service level of a bus transit system and increase bus transit attractiveness. Different
from the previous LRPs [14]-[19], we suppose that a lane on a road segment can be reserved only when the bus volume per unit time on it reaches a given volume level. This assumption attempts to maximize bus exclusive lanes effectiveness. The objective of a BLRP is to minimize the negative traffic of reserved lanes. We first formulate the problem as an integer linear programming model, and show that its complexity is NP-hard. We then propose a cut-and-solve based algorithm to exactly solve the model. Computational results on randomly generated instances show that the proposed cut-and-solve algorithm is more efficient than the well-known optimization software CPLEX 12.4 [20]. The remainder of the paper is constructed as follows. In Section II, we first describe the BLRP and formulate an ILP model for the considered problem. Then, we analyze its complexity. Section III sketches an optimal method based on cut-and-solve algorithm. Computational results are presented in Section VI. This work is concluded in Section V. II. FORMULATION AND COMPLEXITY ANALYSIS A. Description and Formulation TABLE I NOTATIONS FOR BUS LANE RESERVATION PROBLEM
Input parameters N: set of nodes in graph G A: set of arcs in graph G K: set of bus routes, kK Ta: travel time of buses on arc a if no lane is reserved, a A T′a: travel time of buses on arc a if a bus exclusive lane exists, a A Tk: given deadline for the k-th bus route, kK Ca: traffic impact if a lane is reserved for buses on arc a, a A fk: number of buses on the k-th bus route per unit of time, kK , which reflects the frequency of service. Da: given bus volume per unit time for reserving a lane on arc a, a A k k S a : status if bus lane k passes arc a or not, and S a =1 or 0. The former denotes bus route k passes arc a, and the later denotes it does not, a A. Decision variable Za: Za=1, if a lane on arc a is reserved; and otherwise Za=0, a A. Let G = (N, A) denote an urban bus transportation network in which N is a set of nodes and A is a set of directed arcs. A node and an arc represent a road intersection and a road segment, respectively. For simplification, bus stops are placed on nodes like that in [11]. Let k, K, and |K| be the k-th bus route, a set of bus routes and the number of bus routes, respectively. We have kK. In addition, the following assumptions are made: 1) the transit bus routes and stop locations are predetermined and there are at least two lanes on each arc; 2) road segment travel time can be decreased on a reserved lane; 3) each bus route can be composed of GPLs and reserved lanes, i.e., the bus routes can be partially reserved. The goal of
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BLRP is to minimize the negative traffic impact of reserved lanes. The parameters and decision variable used for the formulation is given as Table I. Then, a bus lane reservation problem is formulated as an integer linear programming model as follows:
C
P: Min
defined as follows: there are n items that have to be packed in a knapsack, each item j has an associated profit pj and weight wj, given a knapsack with capacity c, the objective is to maximize the total profit. Its integer linear programming model can be constructed as follows: n
a
(1)
Za
0-1 KP: Max
S
s.t.
(TaZ a Ta (1 Z a )) Tk ， k K
k a
(2)
n
s.t.
a A |K |
S
k a
(3)
fk Z a Da , a A
(4)
Z a {0,1}, a A
The objective function (1) is to minimize total reserved lanes’ negative traffic impact. Constraint (2) represents that the total travel time of the k-th bus route should not exceed its given travel deadline. Constraint (3) guarantees that the bus volume on arc a should exceed a certain level if it is reserved. Constraint (4) is binary constraint on the decision valuables. Let A1 denote the set of arcs on which the bus volume does |K |
not exceed a certain level, i.e., for aA1,
S
k a
f k D a . Thus,
k 1
the proposed model P can be easily simplified as the following model P′.
C
Za
a
a A
s.t. Constraints (2) and (4) (5)
Z a 0, a A1
B. Complexity of BLRP Theorem 1: A BLRP is a NP-hard problem Proof: First, we will show that the special case of a BLRP, where there is only one bus route, i.e., |K|=1, and Da is small enough such that constraint (3) can be relaxed, is NP-hard. The special case of the BLRP (we call it SBLRP) that can be represented by the following program. SIP: Min
C
a
Za
a A
s.t.
(T Z a
a
Ta (1 Z a )) TC
(6)
a A
Z a {0,1}, a A
(7) where A denotes the set of arcs on the only bus route, and TC denotes a given travel deadline. Let |A| denote the number of arcs in A. SIP can be easily transformed to the following equivalent form. SIP′: Min
C
a
Za
a A | A |
s.t.
(T a 1
| A |
a
T a ) Z a
T
a
TC
(8)
a 1
Z a {0,1}, a =1, 2, ..., A
(9)
Then, we demonstrate the NP-hardness of SIP′ through a reduction from the 0-1 Knapsack Problem, which is known to be NP-hard [21]. The 0-1 Knapsack Problem (0-1 KP) is
w
j
(11)
xj c
j 1
(12)
x j {0,1}, j 1, 2, ..., n
k 1
P′: Min
(10)
pjxj
j 1
a A
Finally, we explain how to transform 0-1 KP to SBLRP. First, n should correspond to |A|, j (j = 1, 2, …, n), and pa, wa, | A |
and c should be correspond to Ca, Ta -T′a, T C T a , a 1
respectively. Second, xa should be replaced by 1-Za. Note that after completing the two steps above, a 0-1 KP is transformed to a SBLRP, that is to say, 0-1 KP is reducible to SBLRP. As analyzed above, 0-1 Knapsack Problem that is known to be NP-hard [21] is reducible to a special case of the BLRP, i.e., SBLRP. Hence, a BLRP is also NP-hard. III. SOLUTION METHOD In this paper, we aim to propose an exact method to solve the model P′. As previously mentioned, several cut-and-solve algorithms were successfully proposed to exactly solve LRPs [16]-[18]. This inspires us to apply a cut-and-solve algorithm to exactly solve the new variant of LRP. The cut-and-solve algorithm was first introduced for the traveling salesman problem (TSP) [22]. Later, Yang et al. [23] applied it to solve facility location problems. Its main part is cut-and-solve iterative. We simply explain its procedure as follows. A piercing cut (Pn) is generated at the n-th iteration (n 1) to partition the solution space of the current problem (CPn) (CP1 is defined as the original problem) into two subspaces. They correspond to two subproblems: sparse problem (SPn) and dense problem (DPn), respectively. Due to the sparse solution space, an optimal solution of SPn can be easily found by an IP solver, such as CPLEX’s. The optimal objective value of SPn is an upper bound of the original problem (minimization problem), denoted by UBn. The current best upper bound denoted by UBb is updated if UBn is smaller than UBb. A lower bound of the origin problem denoted by LBn can be found by solving the linear relaxed problem of DPn. Obviously, if UBb is smaller than LBn, UBb is the optimal objective value and the iteration stops. Otherwise, CPn+1 is defined as DPn and a new iteration starts. We refer readers to [22] for more details. For a cut-and-solve algorithm, an appropriate Pn is critical. It combines a set of decision variables, denoted by Un. Un is defined according to the reduced costs of valuables in [22], which is composed of valuables whose reduced costs are greater than a given positive value, denoted as hn. Due to that the variables in Un are all binary, the sum of them is either greater than 1 or equal to 0. With such a piercing cut, CPn is separated into DPn (with the sum of all valuables in Un is
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greater than1) and SPn (with the sum of all valuables in Un is equal to 0). As demonstrated in [22], the way defining Pn above achieved good performance on TSP. Fang et al. [16], [17] also adopted such a Pn for LRPs. Moreover, the objective of the BLRP is directly related with valuables Za, aA, like that in TSP [22]. This inspires us to use reduced costs of variables to define the piercing cut. Therefore, Un is defined as follows: Un = {Za | (Za) > hn, aA}
Theorem 2: For n2, if U′1 U′2 … U′n-1 U′n holds, SPn and DPn are equal to SPn and DPn, respectively. Proof: Its correctness can be proved similar to Theorem 3 in [16], for more details, please see [16]. The overall algorithm for the BLRP is summarized as Algorithm 1. Algorithm 1: Cut-and-solve for the BLRP 1: Initialize n=1, UBb =+, and CP1 = P′. 2: Solve CPn’s linear relaxation problem to obtain the reduced costs of all variables. 3. Define Pn by (18) and obtain SPn and DPn. 4. Solve SPn exactly to obtain UBn. If UBn < UBb, UBb= UBn. 5: Solve the linear relaxation problem of DPn to obtain LBn. If LBn UBb, output UBb and the corresponding solution as the optimal objective function value and the optimal solution, respectively, and end; 6. Set CPn+1 = DPn, n = n +1, and go back to Step 2.
(13)
where (Za) is the reduced cost of Za, and the value of hn is set as 0.1×max{ (Za)| aA} like that in Fang et al. [17]. The reduced costs of variables Za, aA, are obtained by solving the linear relaxation problem of CPn. The Pn is defined as follows:
(Pn)
(14)
Za 1
Z a U n
With Pn, two subproblems SPn and DPn are defined as follows, respectively. (SPn) Min
C
a
Za
a A
s.t. Constraints (2), (4), and (5)
Z a 1, t 1, 2, ..., n 1
(15)
Za =0
(16)
Z a U t
Z a U n
(DPn) Min
C
a
Za
a A
s.t. Constraints (2), (4), (5), (14) and (15) In order to further improve the algorithm’s performance, the improved piercing cut P′n proposed by Fang et al. [16] based on the basis of Pn is adapted here. A new definition of Un (n 2) used for the definition of P′n, denoted as U′n, is given as follows: U′n = {Za | (Za) > hn, Za U′n-1, aA}
(17)
When n = 1, U′1= U1.
(P′n)
(18)
Za 1
Z a U n
With P′n, two new subproblems SPn (n 2) and DPn (n 2) can be defined as follows: (SPn): Min C a Z a a A
s.t. Constraints (2), (4), and (5)
(19)
Za 1
Z a U n -1 \ U n
(20)
Za =0
Z a U n
(DPn) Min
C
a
Za
IV. COMPUTATIONAL RESULTS This section reports experiment computational results on 15 randomly generated problems sets with five instances in each, i.e., 75 instances in total, tested on a personal computer with 2.5 GHz CPU and 2.95 GB RAM. The proposed method is coded in C++ embedded with the optimization software CPLEX (version 12.4) (CPLEX 12.4 in short) in default setting used to optimally solve SPn and the relaxation problem of DPn. Its performance is evaluated by comparing computation time (CPU time in seconds) with CPLEX 12.4. The instances are randomly generated as follows. The graph G = (N, A) is generated based on the network model proposed by [24]. The existence of an arc for a pair of nodes depends on a probability function associated with the two nodes’ distances. The travel time Ta on an arc a if no lane is reserved is defined as La/Pa, where La and Pa are the distance, the average travel speed on arc a without bus exclusive lane, respectively. The travel time on a bus exclusive lane Ta is defined as baTa, where ba is randomly generated in [0.5, 0.8] like those in [14]-[19]. The impact parameter of a reserved lane Ca is defined as daTa according to [14], which corresponds to the increased time caused by a reserved lane, where da is randomly generated in [0.2, 0.3]. To avoid a trivial problem, the deadline Tk is randomly and uniformly generated between the total travel time on an entire reserved bus path and on a non-reserved path. Integer fk is randomly generated in [5, 12], which means that bus service frequency ranges from 5 to 12 minutes, i.e., 5 to 12 vehicles/hour. As pointed out by Seo et al. [13], a reserved lane with a bus volume in the range of 20-400 vehicles/ hour is useful, so Da is set as 20vehicles/hour, which is the lowest level. Let CT, and CTC denote the computation time spent by our algorithm and CPLEX, respectively. The computational results are summarized in Tables II and III, and Fig.’s 1 and 2.
a A
s.t. Constraints(2), (4), (5), and (18) When n = 1, SP1 = SP1, and DP1 = DP1.
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TABLE II COMPUTATIONAL RESULTS FOR INSTANCES WITH 20 FIXED NUMBER OF BUS ROUTES |K| Set
|N|
|K|
CT (s)
CTC(s)
CT/ CTC
1
50
20
0.37
0.39
0.94
2
60
20
1.19
1.50
0.79
3
70
20
1.41
1.62
0.87
4
80
20
1.75
2.07
0.85
5
90
20
3.57
4.20
0.85
1.66
1.96
0.85
AVG.
can find that CT is less than CTC over all the nine sets 6-14 in Table III. In addition, it can be seen in Table III that CT/ CTC ranges between 0.64 and 0.87, and its average value for all the sets is 0.73. This means that the proposed algorithm can find optimal solution by spending 73% the computation time of that spent by CPLEX for all the instances on the average. This indicates that the proposed algorithm is more efficient than CPLEX. TABLE III COMPUTATIONAL RESULTS FOR INSTANCES WITH |K| INCREASING FROM 20 TO 60
Table II reports the computational results for instances with 20 fixed number of bus routes and number of nodes increasing from 50 to 90. We can see from Table II that both the proposed algorithm and CPLEX can exactly solve all instances within relatively short computation time. The computation time by the proposed algorithm, i.e., CT, is less than that spent by CPLEX, i.e., CTC, for all the five sets 1-5. CTC increases from 0.39s to 4.2s, and its average value is 1.96s, whereas the CT increases from 0.37s to 3.57s, and its average value is 1.66s. We can also see that the value of ratio CT/ CTC changes between 0.79 and 0.94, and its average value is 0.85. This indicates that the proposed algorithm only spends 85% average computation time of that spent by CPLEX. This shows that the proposed algorithm is better than CPLEX.
Set
|N|
|K|
6
100
20
4.07
5.56
0.73
7
100
25
14.57
16.68
0.87
8
200
30
20.52
24.44
0.84
9
200
35
197.28
236.57
0.83
10
300
40
228.11
286.11
0.80
11
300
45
379.84
471.16
0.81
12
400
50
537.99
841.95
0.64
13
400
55
1182.38
1761.56
0.67
14
500
60
1438.45
1877.62
0.77
444.80
613.52
0.73
AVG.
CT(s)
CTC(s)
CT/ CTC
Fig. 2 Computational results for instances with |K| increasing from 20 to 60
Fig. 1 Computational results for instances with 20 fixed number of bus routes
In addition, it can be seen from Table II and Fig. 1 that CT and CTC both increase with the number of nodes |N|, while CT increases more gradually than CTC. This also indicates that the proposed algorithm is more efficient than CPLEX for instances with fixed number of bus routes |K| and varying number of nodes |N|. Table III presents the comparison results of the proposed algorithm and CPLEX for instances with |N| increasing from 100 to 500 and |K| varying from 20 to 60. It can be found from Table III that both methods can obtain optimal solutions for all instances in 1900s. The computation time spent by CPLEX increases from 5.56s to 1877.62s and its average value for all instances is 613.52s, whereas the computation time spent by the proposed cut-and-solve algorithm increases from 4.07s to 1438.45s and its average value for all instances is 444.8s. We
Moreover, from Table III and Fig. 2, we can find that the computation time by our proposed algorithm and CPLEX both increases with the number of bus routes |K|, but CT increases more gradually than CTC from sets 6-11. Note that CT increases a bit faster than CTC from sets. This may because that the solution space for these larger-size problem sets is larger and the generated piercing cuts in our cut-and-solve algorithm may be not very efficient that influences the convergence of the algorithm. Although so, it can be found that the computation time spent by the proposed algorithm is much less than that spent by CPLEX for the larger-size problem sets 12-14. V. CONCLUSION This work investigated a new variant of lane reservation problem called a bus lane reservation problem (BLRP). The problem aims to optimally choose some existing general lanes
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in a predetermined bus transit network and convert them into bus exclusive lanes via lane reservation so that the total travel time of buses on each bus route is less than a given deadline. This aims to improve the service level of a bus transit system. Meanwhile, the bus volume on a reserved lane should exceed a certain bus volume level to maximize its effectiveness. The bus lane reservation problem aims to minimize the impact caused by reserved lanes. An integer linear programming model is formulated and it is demonstrated to be NP-hard. Then, a cut-and-solve algorithm is proposed to optimally solve the BLRP. The computational results on randomly generated instances show its efficiency. The considered problem in this paper is the first work for the bus lane reservation problem that can serve as a basic one to study more complicated problems. In the future, on one hand we will study a more realistic bus lane reservation problem considering more realistic factors such as bus transit demand, travel time windows for bus stops, and optimal selection of paths between two consecutive bus stops. In addition, the evaluation of impact caused by reserved bus lanes is very complex and it needs more studies. On the other hand we will try to analyze characteristics of the BLRP for the proposed algorithm such that the computation speed of the algorithm can be accelerated. Besides, although the proposed algorithm can solve larger-size instances compared to the studies in the literature, in real life the network size and the number of bus routes are very large. Exact methods are difficult to exactly solve instances of realistic size within acceptable computation time due to their NP-hard nature. Thus, effective and efficient heuristics or metaheuristics, e.g., [25] -[28] should be developed for large-size problems. REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9]
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