A parallel EAX-based algorithm for minimizing the number of routes in the vehicle routing problem with time windows
Mirosław Błocho∗†
∗Silesia University of Technology Gliwice, Poland
†ABB ISDC Krakow, Poland
email: Miroslaw.Blocho@pl.abb.com
Zbigniew J. Czech∗†
∗Silesia University of Technology Gliwice, Poland
†Silesia University Sosnowiec, Poland email: Zbigniew.Czech@polsl.pl
Abstract—A parallel EAX-based algorithm for minimizing the number of routes in the vehicle routing problem with time windows is presented. The main contribution is a novel approach concerning the usage of the edge assembly crossover (EAX) operator during exchanging the best solutions between the processes. The objective of the work is to analyze the speedup, achieved accuracy of solutions and scalability of the MPI implementation. For comparisons the selected Gehring and Homberger’s (GH) tests are used. The results of the extensive computational experiments indicate that the new algorithm based on the EAX approach is not only very fast but also very effective. The eight new best-known solutions for the GH benchmarking tests were found by making use of the algorithm.
Keywords-parallel algorithm, vehicle routing problem with time windows, guided local search, metaheuristics, EAX oper- ator, MPI library
I. INTRODUCTION
The vehicle routing problem with time windows (VRPTW) can be described as a problem of determining a least cost scheduling plan to deliver goods from a single depot to a set of geographically scattered customers. Each customer must be visited exactly once by a single vehicle, which departs from and returns to the depot. Additionally, the total of goods loaded in any vehicle must not exceed the maximum vehicle capacity (capacity constraint) and each customer must be serviced within a given time interval (time window constraint) [31].
The VRPTW belongs to a large family of the vehicle routing problems which have numerous practical applica- tions such as: airline fleet routing, rail distribution etc. [14].
The VRPTW is also NP-hard in strong sense as it includes several NP-hard optimisation problems such as TSP and bin packing [19], [32].
Because of practical importance and wide applicability of the VRPTW, many algorithms have been developed so far for this combinatorial optimization problem. The con- struction heuristics, improvement heuristics (so-called local searches) and metaheuristics are among the most successful algorithms to solve the VRPTW.
The construction heuristics are based on generating a fea- sible solution by iterative inserting customers into partial routes according to specified criteria such as minimum additional travel distance or maximum savings. During con- struction process the feasibility of partial solutions must not be violated at anytime. The examples of these heuristics were successfully applied by Mester [18], Solomon [31], Dullaert and Br¨aysy [8] and Potvin and Rousseau [26].
In turn, the improvement heuristics modify an actual solution by executing consecutive local search moves in order to find better neighboring solutions. The neighborhood includes solutions, which can be obtained from the current one by exchanging edges or customers. The examples of local search heuristics can be found in Russell [29], Br¨aysy [4], Caseau and Laburthe [5], Potvin and Rousseau [25] and Thompson and Psaraftis [33].
The metaheuristics belong to a group of most powerful algorithms, which are based on exploring the solution space for finding better solutions and they usually embed im- provement heuristics as well as standard route construction algorithms. The metaheuristics allow for infeasible partial solutions and for a deteriorating of solutions in the search process in order to escape local minima. The highest perfor- mance for the VRPTW has been obtained by the application of genetic algorithms [15], evolution strategies [27] and tabu searches [13]. The genetic algorithms were successfully applied by Berger et al. [1] and Jung and Moon [16]. In turn, the tabu searches with outstanding results were applicated by Taillard et al. [32], Chiang and Russell [6], Rochat and Taillard [28], Cordeau et al. [7] and Schulze and Fahle [30].
The metaheuristics with evolution strategies were applied by Mester [19] and Gehring and Homberger [10], [11], [14].
This paper presents a parallel EAX-based algorithm for minimizing the number of routes in the vehicle routing problem with time windows. It extends our previous re- search concerning the parallel heuristic (BC) for the route- minimization stage in the VRPTW [2], [3] and Nalepa and Czech’s improvements [20]. The parallel BC algorithm exhibited very good performance in terms of quality and
1: for Pi← P1 to PN do in parallel
2: create the initial solution σ
3: while not finished do
4: initialize the ejection pool (EP ) with the customers from a randomly removed route
5: initialize the penalty counters p[vi] ← 1(i = 1, ..., M )
6: while EP 6= ∅ and not f inished do
7: cre ← cre+ 1 . increase elimination steps count
8: select and eject the customer vi from EP
9: if Sinsertf e (vi, σ) 6= ∅ then
10: σ ← a new solution σ0 selected randomly from Sinsertf e (vi)
11: else
12: σ ← Squeeze(vi, σ)
13: end if
14: if vi is not inserted into σ then
15: p[vi] ← p[vi] + 1
16: σ ← σ0 from Sej(vi) with min. Psum = p[vout(1)] + p[vout(2)] + ... + p[vout(km)]
17: add the ejected customers: vout(1), v(2)out, ..., vout(km)to EP
18: σ ← P erturb(σ)
19: end if
20: if cre mod δc = 0 then
21: f inished ← Cooperation(σ) . EAX recombination performed during co-operation
22: end if
23: end while
24: end while
25: end for
Figure 1. Parallel EAX-based algorithm
accuracy of solutions and enabled us to find the twelve new best-known solutions. It originated from the route minimiza- tion algorithm by Nagata and Br¨aysy [21] and was based on the notion of the ejection pool used in the heuristic by Lim and Zhang [17], combined with guided local searches and the diversification strategy by Voudouris and Tsang [34]. In this work, we suggest a novel approach based on usage of the edge assembly crossover (EAX) operator during exchanging best solutions between the processes. The EAX operator for the VRPTW was first proposed by Nagata et al. [22]. Its basic idea is to remove and replace edges in two selected parent solutions, considering common and different edges.
The objective of the work is to analyze the achieved accuracy of solutions, and the speedup and scalability of the MPI implementation. The parallel EAX-based algorithm is presented in section II. Section III contains the discussion of the experimental results. Section IV concludes the paper.
II. PARALLELEAX-BASED ALGORITHM
The parallel EAX-based algorithm for minimizing the number of routes in the VRPTW extends the parallel version of the algorithm by Błocho and Czech [3] by introducing the usage of the EAX operator during the phases of exchanging the best solutions between the processes.
The parallel algorithm contains N components which are
executed as processes P1, P2, . . . , PN (Fig. 2). The algo- rithm starts with an initial solution σ where each customer is served by a single vehicle (line 2 in Fig. 1). The consecutive attempts to reduce the number of routes in σ are then applied until the total computation time reaches a given limit. A randomly selected route is removed from σ and the excluded customers are used to initialize the ejection pool EP(line 4). The penalty counters p[vi] for all customers are set to 1 (line 5), whose role is to estimate the difficulties of re-inserting the customers back into σ.
In each iteration (lines 6-23) a customer is selected from EP and the attempts are made to insert it into the routes of σ without violating the capacity and time windows constraints. At first, a set containing all feasible insertion positions Sinsertf e (vi, σ) is considered (line 9). If there exists at least one feasible solution then it is chosen randomly from this set (line 10), otherwise the function Squeeze is called (line 12), which temporarily accepts an infeasible insertion with the minimal penalty function value Fp(σ) defined in [21]. A solution with the smallest penalty is selected and the local search moves are undertaken in order to restore the feasibility.
If the squeezing could not produce a feasible solution then the penalty counter of the customer is increased (line 15) and the ejections of other customers are tested (at most
σ(init)1 ⇒ σ2(init) ⇒ σ3(init) ⇒ • • • ⇒ σN −1(init) ⇒ σ(init)N ⇒ σ(init)1
↓ ↓ ↓ ↓ ↓ ↓
σ1(1δc) ⇒ σ(1δr c)
1(1) ⇒ σr(1δc)
1(2) ⇒ • • • ⇒ σr(1δc)
1(N −2) ⇒ σ(1δr c)
1(N −1) ⇒ σ1(1δc)
↓ ↓ ↓ ↓ ↓ ↓
σ1(2δc) ⇒ σ(2δr c)
2(1) ⇒ σr(2δc)
2(2) ⇒ • • • ⇒ σr(2δc)
2(N −2) ⇒ σ(2δr c)
2(N −1) ⇒ σ1(2δc)
↓ ↓ ↓ ↓ ↓ ↓
• • • • • • • • •
↓ ↓ ↓ ↓ ↓ ↓
σ1(xδc) ⇒ σr(xδc)
x(1) ⇒ σ(xδr c)
x(2) ⇒ • • • ⇒ σ(xδr c)
x(N −2) ⇒ σr(xδc)
x(N −1) ⇒ σ1(xδc)
Figure 2. The co-operation scheme of N processes (σ1- solution of the master process P1; σi(init)- initial solution of the process Pi; σr(jδc)
j(i) - solution of the process Prj(i)at j co-operation step; rj(i) - i element of a random permutation rj ∈ {2, . . . , N } at j co-operation step; x - co-operation steps count; δc- number of steps between co-operations)
km customers [21]). For this purpose the set Sej(vi) is generated, which contains the solutions with various ejected customers and the given customer inserted at different route positions. The customers with minimized sum of the penalty counters are ejected (line 16) and added to EP (line 17) as they should be relatively easy to reinsert in further steps.
The obtained solution is then perturbed with a number of feasible local moves (line 18).
The processes co-operate periodically every δcelimination steps (line 20; smaller δc means higher frequency of co- operation). At the beginning of each co-operation step the master1process P1generates a random sequence of the pro- cesses rj ∈ {2, . . . , N } and broadcasts it asynchronosuly.
Such a co-operation scheme differs from our previous more static approach [3] and allows for increasing the chances of better reproductive opportunities. The scheme of the co- operation between the processes is illustared in Fig. 2. Every co-operation step is started by P1 and the best solution is transferred to the rest of the processes until it reaches the last one in the actual permutation rj. Then the best achieved solution is sent back to the master process P1. Such a way of the co-operation between the processes guarantees that (1) P1 holds the best solution found to date, (2) all other processes hold at least as good solution as the best found in a previous co-operation step. Furthermore, in order to achieve a sufficient diversification, the processes create their own solutions with a number of route elemination steps and then the co-operation is started.
If a process receives a solution with a smaller number of routes then it replaces the current one. Otherwise, the EAX operation is performed on both solutions and a new children solution σch is created. Then the local moves are carried out to restore the feasibility of σch(if it is not already feasible) due to the criteria based on decreasing the sum of
1We distinguish process P1 calling it master because it decides on the order in which processes co-operate in each phase. Moreover, it controls the execution time of the algorithm by sending to the remaining processes the signal to finish computations when this time elapsed.
the capacity and time window penalties in each repairing iteration. If σchbecomes feasible then it replaces the current solution. This approach is motivated by the efficiency of hy- brid genetic algorithms, combining evolutionary adaptation of a population with individual learning within a lifetime.
The master process P1 determines the execution time of the algorithm by sending in each co-operation step (line 21) to the remaining processes the signal to finish computations when this time elapsed.
The edge assembly crossover (EAX) operator was first proposed for the travelling salesman problem (TSP) by Na- gata and Kobayashi [23] and later adapted to the capacitated vehicle routing problem (CVRP) by Nagata [24] and finally to the VRPTW by Nagata [21]. The main idea of EAX is to remove and replace edges in a chosen parent solution with the other parent solution, based on the common and different edges. The EAX operator generates each time only one offspring solution and consists of five main steps, as illustrated in Fig. 3.
In step I, a graph GAB = (GA∪ GB)\(GA∩ GB) is formed, where GAand GBcorrespond to the graphs created from the parent solutions σAand σB. In other words, graph GAB contains the edges by which the GA and GB graphs differ from each other.
In step II, all edges from GAB are divided into so-called AB-cycles. An AB-cycle is defined as a cycle on GAB such that edges from σA and σB are linked alternately in the opposite directions. The process of creating AB-cycles is started by randomly selecting an initial node on GAB and then tracing in turn edges belonging to σA and σB until a cycle is found. Then the AB-cycle is removed from GAB
and the process of generating AB-cycles is continued until all edges in GAB have been removed (as an example, consider Fig. 3 where six AB-cycles were found).
In step III, an E-set is formed by a combination of any AB-cycles. Two strategies called single and block can be applied (in the example of Fig. 3, a single strategy is used to select the third AB-cycle as the E-set).
Figure 3. The illustration of the EAX operator
In step IV, σA is chosen as the base solution and an in- termediate solution is constructed by removing E-set ∩ GA
and adding E-set ∩ GB.
In the last V step, the possible subtours are merged with the routes already connected to the depot.
III. EXPERIMENTAL RESULTS
The algorithm was implemented in C++ using the MPI interface and was tested on the benchmarking tests proposed by Gehring and Homberger [12]. The computations were performed on Galera supercomputer [9] on nodes equipped with Intel Xeon Quad Core 2.33 GHz processors, each with 12 MB level 3 cache. The nodes were connected by the Infiniband DDR fat-free network (throughput 20 Gbps, delay 5 µs). The computer was executing Linux operating system and the source code was compiled using Intel 10.1 compiler and MVAPICH1 v. 0.9.9 MPI library. The maximal execution time was set to 30 minutes, the number of steps between co-operations δc was set to 100, the maximal number of customers to be ejected km was set to 4 and the EAX strategy (single, block) was chosen randomly.
In order to evaluate the MPI implementation of the parallel algorithm, it was tested on 300 benchmarking tests elaborated by Gehring and Homberger [12]. Six sets of test cases (C1, C2, R1, R2, RC1, RC2) were designed to highlight the factors that affect a behavior of the routing algorithms including the geographical data, the average
number of customers serviced by a vehicle, the positioning and tightness of the time windows. The customer locations are randomly generated in sets R1 and R2, clustered in sets C1 and C2, and a mix of clustered and random locations in sets RC1 and RC2. The sets C1, R1 and RC1 have a short scheduling horizon and allow for only a few customers per route. In contrast, the sets C2, R2 and RC2 have a long scheduling horizon, which permits many customers to be serviced by the same vehicle.
Overall 15000 experiments (50 for each test instance) were carried out. The results obtained through the extensive tests showed that the EAX-based algorithm (called further EAX) was competitive to the existing basic parallel approach [3] (called further BC). The testing results of the implemen- tation on 64 processors2indicate that 89% of GH tests were solved to good accuracy in time less than 10 seconds and only benchmarking tests from C/RC groups having 600-1000 customers were more difficult to solve in such a short time.
Based on the experimental results, the average number of ejected customers ranged between 1 and 2 for C tests and between 1 and 3 for R/RC tests. The ejections of km = 4 customers were tested only in several R test instances having 800-1000 customers. The number of steps between co-operations set to δc = 100 proved to be a good choice for most of the C/RC tests. However smaller value of δc
2Each process was run on a single processor
Table I
NEW BEST-KNOWN SOLUTIONS(N –CUSTOMERS COUNT, p – PROCESSES COUNT)
Best-known New best
Instance N p number number Time [min]
of routes of routes
RC2 4 5 400 16 9 8 1
C1 6 6 600 64 60 59 19
C1 8 2 800 16 73 72 0.5
C1 8 6 800 64 80 79 15
C2 8 6 800 64 24 23 29
C1 10 6 1000 64 29 28 16
C1 10 7 1000 64 98 97 28
C2 10 8 1000 64 100 99 18
probably could give better results for R tests where more extensive search process was required and higher frequency of co-operation would be very beneficial.
Considering the best solutions found, the EAX-based algorithm outperforms the BC algorithm by finding 8 new best-known solutions (see Tab. I) and 287 ties. Moreover, comparing the CVN (Cumulative Vehicles Number) of all GH tests, the EAX-based algorithm proved to be much more effective than BC one by obtaining CVNEAX = 10295 instead of CVNBC = 10304. It is worth noting that the CVN of all best world results published so far by Sintef [12] is also higher (CVNworld = 10297) than the value obtained by the proposed EAX-based algorithm. The detailed comparison of the CVN values between the EAX-algorithm and the world- best results is given in Tab. II.
A number of GH tests were selected to analyze the speedup of the EAX-based implementation. The results of the experiments are illustrated in Fig. 4 and additionally compared with the results obtained by the BC algorithm.
The detailed comparisons of the average execution times along with the achieved speedups are presented in Tab. III (C2 6 2, C2 8 9, C2 10 7) and in Tab. IV (RC2 6 2, RC2 8 5, RC2 10 5).
It can be noticed that the EAX-based algorithm obtains better speedups than the BC one, for the larger number of processes (≥ 16). The achieved speedups for the tests from RC groups for the EAX-based algorithm are slightly smaller than speedups for tests from C groups (what is not a case for the BC algorithm). It may indicate that the benefits coming from the recombination of solutions by the EAX operator exceed the overhead associated with processes communication, idling, synchronization and EAX operations. The better speedups obtained for C tests can be explained by the fact that the EAX operator generates offspring solutions without introducing long edges, which is particularly important for the clustered customers. The new best-known solutions were found mostly for the tests from C groups (7 out of 8), thus additionally confirming the above
Table III
RESULTS FOR SELECTEDGHTESTS: C2 6 2, C2 8 9, C2 10 7 (p –NUMBER OF PROCESSES, ¯τ –AVERAGE EXECUTION TIME[S],
S –SPEEDUP)
p ¯τ (EAX) τ (BC)¯ S(EAX) S(BC)
C2 6 2
1 492.1 492.1 1.00 1.00
8 60.8 65.4 8.09 7.52
16 34.6 49.2 14.21 10.01
24 23.6 28.1 20.85 17.54
32 16.7 19.8 29.41 24.92
40 14.0 16.0 35.12 30.77
48 11.7 14.3 42.15 34.48
56 10.4 12.6 47.20 39.18
64 9.3 11.0 52.69 44.66
C2 8 9
1 1564.6 1564.6 1.00 1.00
8 211.1 213.7 7.41 7.32
16 112.5 152.5 13.91 10.27
24 77.3 90.9 20.23 17.22
32 55.8 64.6 28.03 24.21
40 48.6 54.6 32.19 28.64
48 39.3 49.2 39.77 31.82
56 34.5 42.5 45.30 36.81
64 29.6 36.5 52.90 42.92
C2 10 7
1 852.7 852.7 1.00 1.00
8 121.6 122.5 7.01 6.96
16 71.2 90.1 11.97 9.47
24 41.0 53.1 20.81 16.06
32 31.7 40.6 26.92 21.02
40 26.5 31.6 32.14 27.00
48 21.9 26.1 38.87 32.70
56 19.8 24.3 42.97 35.08
64 17.2 21.4 49.48 39.92
explanation.
Similarly as in [20], the problem of saturating the popu- lation of solutions occured also in our algorithm. Especially in case of experiments with a small number of processes (2 to 8) this effect was particularly perceptible and the gain from usage of the EAX operator was hardly noticeable.
The probability of ending up with a set of similar solu- tions is much smaller in case of large populations and for that reason our algorithm obtained much better results for larger number of processes (≥ 16), as indicated in Fig. 4.
Generating a random permutation of processor numbers in each co-operation as well as allowing for a co-operation only after the base solutions (with a specified number of
1 16 32 48 64
1 16 32 48 64
S
p
C2 6 2 Ideal
EAX
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◦ BC
N N N N N N N N N N N N N N N N N N
1 16 32 48 64
1 16 32 48 64
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◦ BC
N N N N N N N N
N N N N N N N N N N
1 16 32 48 64
1 16 32 48 64
S
p
C2 8 9 Ideal
EAX
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C2 10 7 Ideal
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N N N N N NNN N N N N N N N N N N
1 16 32 48 64
1 16 32 48 64
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p
RC2 10 5 Ideal
EAX
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N N N N N N N N N N
Figure 4. Speedup S vs. number of processes p for tests C2 6 2, C2 8 9, C2 10 7, RC2 6 2, RC2 8 5 and RC2 10 5 (Ideal - theoretical speedup, EAX - speedup of the parallel EAX-based algorithm, BC - speedup of the parallel algorithm [3])
route elimination steps) have already been created by each process, seem to be justified. These features of our algorithm significantly help to achieve a sufficient diversification of the solutions, thus minimize the risk of saturating the population and increase the chances of better reproductive opportunities.
IV. CONCLUSIONS
The parallel EAX-based algorithm for minimizing the number of routes in the VRPTW along with the experimental results of the MPI implementation were presented. The new approach differs from the previous parallel algorithm in the way the recombination of the solutions is performed. For this goal the EAX operator is used. The new algorithm proved very robust and effective by solving nearly 100% of GH
problem instances to the current known optimum and finding new best-known solutions to eight tests, with respect to the number of routes. The extensive experiments for various benchmarking tests showed that the relative speedups were significantly larger than those achieved by the BC parallel algorithm.
Solving the VRPTW is based on a two-stage approach, which allows for joining the presented algorithm with other heuristics, e.g. tabu search or genetic algorithms. Our ongoing research is focused on the parallelization of the edge assembly memetic algorithm by Nagata et al. [22]
and combining it with the presented EAX-based parallel algorithm for minimizing the number of routes in the first stage of the VRPTW.
Table II
COMPARISON OFCVN (CUMULATIVEVEHICLENUMBER)VALUES FORGHTESTS BETWEENEAX-ALGORITHM AND WORLD-BEST PUBLISHED RESULTS[12] (BOLDFACED RESULTS WITH BETTERCVN).
C1 C2 R1 R2 RC1 RC2 CVN
EAX/best EAX/best EAX/best EAX/best EAX/best EAX/best EAX/best
200 189 / 188 60/60 182 / 181 40 / 40 180 / 180 43 / 43 694 / 692 400 376 / 376 117 / 117 364 / 362 80 / 80 360 / 360 84 / 85 1381 / 1380 600 573 / 574 174 / 174 545 / 545 110 / 110 550 / 550 114 / 114 2066 / 2067 800 750 / 752 233 / 234 728 / 727 150 / 150 720 / 720 154 / 154 2735 / 2737 1000 940 / 941 288 / 289 919 / 919 190 / 190 900 / 900 182 / 182 3419 / 3421
CVN 2828 / 2831 872 / 874 2738 / 2734 570 / 570 2710 / 2710 577 / 578 10295 / 10297
ACKNOWLEDGMENT
We thank the following computing centers where the com- putations of our project were carried out: Academic Com- puter Centre in Gda´nsk TASK, Academic Computer Centre CYFRONET AGH, Krak´ow (computing grant 027/2004), Pozna´n Supercomputing and Networking Center, Interdisci- plinary Centre for Mathematical and Computational Model- ing, Warsaw University (computing grant G27-9), Wrocław Centre for Networking and Supercomputing (computing grant 30).
We would like to thank also Jakub Nalepa for his valuable comments and feedback on the contents of this paper.
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Table IV
RESULTS FOR SELECTEDGHTESTS: RC2 6 2, RC2 8 5, RC2 10 5 (p –NUMBER OF PROCESSES, ¯τ –AVERAGE EXECUTION TIME[S],
S –SPEEDUP)
p τ (EAX)¯ ¯τ (BC) S(EAX) S(BC)
RC2 6 2
1 2573.1 2573.1 1.00 1.00
8 381.2 374.0 6.75 6.88
16 184.3 203.9 13.96 12.62
24 121.8 123.9 21.13 20.76
32 86.1 91.1 29.87 28.24
40 70.6 73.9 36.43 34.82
48 59.0 62.8 43.61 40.99
56 53.6 58.5 48.03 44.02
64 49.6 52.4 51.89 49.12
RC2 8 5
1 369.7 369.7 1.00 1.00
8 49.6 51.2 7.45 7.22
16 25.9 30.4 14.27 12.16
24 17.5 18.5 21.18 19.97
32 12.3 13.3 30.02 27.81
40 10.1 11.1 36.50 33.20
48 8.2 9.5 44.90 39.01
56 7.3 8.5 50.49 43.39
64 6.9 7.6 53.54 48.97
RC2 10 5
1 454.7 454.7 1.00 1.00
8 61.5 64.2 7.39 7.08
16 31.3 35.6 14.51 12.64
24 20.7 24.1 21.99 18.88
32 14.8 16.8 30.80 27.02
40 12.3 14.8 37.00 30.81
48 10.8 11.9 41.93 38.21
56 9.0 10.4 50.28 43.91
64 8.8 9.5 51.91 47.88
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