the Number of Routes in the Vehicle Routing Problem with Time Windows
Miroslaw Blocho1,3and Zbigniew J. Czech1,2
1 Silesia University of Technology, Gliwice, Poland Zbigniew.Czech@polsl.pl
2 Silesia University, Sosnowiec, Poland Zbigniew.Czech@polsl.pl
3 ABB ISDC, Krakow, Poland blochom@gmail.com
Abstract. A parallel algorithm for minimizing the number of routes in the vehicle routing problem with time windows (VRPTW) is presented.
The algorithm components cooperate periodically by exchanging their best solutions with the lowest number of routes found to date. The ob- jective of the work is to analyze speedup, achieved accuracy of solutions and scalability of the MPI implementation. For comparisons the selected VRPTW tests are used. The derived results justify the proposed paral- lelization concept. By making use of the parallel algorithm the twelve new best-known solutions for Gehring and Homberger’s benchmarking tests were found.
Keywords: parallel algorithm,vehicle routing problem with time win- dows, guided local search, heuristics, approximation algorithms, MPI library.
1 Introduction
This work presents a parallel algorithm for minimizing the number of routes in the vehicle routing problem with time windows (VRPTW). It extends our previous efforts concerning the improved version of the algorithm [2] originated from the heuristic by Nagata and Br¨aysy [19]. The sequential version of the improved algorithm is based on the notion of the ejection pool used in the heuristic by Lim and Zhang [16], combined with the guided local searches and the diversification strategy [30]. The components of the parallel algorithm cooperate periodically by exchanging their best solutions to the VRPTW.
The VRPTW belongs to a large family of the vehicle routing problems (VRP) which have numerous practical applications such as: the school bus routing, newspapers and mail delivery, armoured car routing, rail distribution, airline fleet routing and many others [29]. Apart from the practical significance, the VRPTW is also NP-hard in the strong sense as it contains several NP-hard optimisation problems such as TSP and bin packing [15].
R. Wyrzykowski et al. (Eds.): PPAM 2011, Part I, LNCS 7203, pp. 255–265, 2012.
Springer-Verlag Berlin Heidelberg 2012c
The current state-of-the-art heuristics for the VRPTW can be divided into improvement heuristics (local searches), construction heuristics and metaheuris- tics. The improvement heuristics change the current solution iteratively by exe- cuting local searches for finding better neighboring solutions. The neighborhood contains the set of solutions that can be obtained from the base solution by ex- changing customers or edges. The most successful applications of the improve- ment heuristics can be found in Br¨aysy [3], Potvin and Rousseau [20], Thompson and Psaraftis [28], Caseau and Laburthe [4], and Russell [24].
The construction heuristics generate a feasible solution by inserting customers iteratively into the partial routes due to specific criteria, such as maximum sav- ings or minimum additional distance never violating the solution feasibility. The examples of such heuristics are described in Solomon [26], Potvin and Rousseau [21], Dullaert and Br¨aysy [8] and Mester [17].
The metaheuristics are based on exploring the solution space in order to iden- tify better solutions and they often contain standard route construction as well as the improvement heuristics. As distinct from the classical approaches, the metaheuristics allow infeasible intermediate solutions and a solution deteriorat- ing in the search process to escape a local minima. The best performance for the VRPTW was obtained by the application of the evolution strategies [22], tabu searches [11] and genetic algorithms [12]. For example the evolution strategies were applied by Gehring and Homberger ([9], [10], [13]) and Mester [18] while the tabu searches with excellent results were applicated by Taillard et al. [27], Schulze and Fahle [25], Chiang and Russell [5], Rochat and Taillard [23] and Cordeau et al. [6]. In turn, the genetic algorithms were successfully applied by Jung and Moon [14] and Berger et al. [1].
The VRPTW is defined on a complete graph G = (V , E) with a set of vertices V = {v0, v1, ..., vN} and a set of edges E = {(vi, vj): vi, vj ∈ V ,i = j}. Node v0
represents a depot and the set of nodes{v1, ..., vN} represents the customers to be serviced. With each node vi ∈ V are associated a non-negative service time si(s0= 0), a non-negative load qi(q0= 0) and a time window [ei, li]. Each edge (vi, vj) contains a non-negative travel time cij and a travel distance dij. A solution to the VRPTW is the set of m routes on graph G such that (1) each route starts and ends at the depot; (2) every customer vi belongs to exactly one route; (3) the total load of each route does not exceed Q; (4) the service at each customer vi begins between ei and li; (5) every vehicle leaves the depot and returns to it between e0and l0; (6) the number of routes and total travelled distance are minimized.
The MPI (Message Passing Interface) implementation of the parallel algo- rithm for minimizing the number of routes in the VRPTW is presented. The objective of the work is to analyze speedup, achieved accuracy of solutions and scalability of this parallel heuristic implementation. The remainder of this paper is arranged as follows. In section 2 the parallel algorithm is presented. Section 3 describes the MPI implementation of the algorithm. Section 4 contains the discussion of the experimental results. Section 5 concludes the paper.
2 Parallel Algorithm
The parallel algorithm for minimizing the number of routes in the VRPTW ex- tends the improved sequential version of the algorithm by Blocho and Czech [2].
The parallel algorithm contains N components which are executed as processes P1, P2, . . . , PN (Alg. 1).
1 for Pi← P1 to PN do in parallel
2 initialize the communication requests
3 create the initial solution σ
4 T C ← f alse // termination condition
5 while not T C do
6 σ ← RemoveRoute(σ)
// cooperation during RemoveRoute function execution
7 end
8 gather solutions from all processes in process P1 and produce the best solution σbest
9 free the communication requests
10 end
Algorithm 1. Parallel algorithm (T C- termination condition)
The framework of the heuristic (Alg. 1) consists of the consecutive remove route steps performed until the total computation time reaches a specified limit Tmax.
The RemoveRoute function (Alg. 2) is started by choosing and removing a route randomly from the current solution σ. The customers from this route are used to initialize the Ejection Pool (EP ) that is used to hold the set of cus- tomers missing from σ. The penalty counters p[vi] initialized at the beginning of the function indicate how many times the attempts to insert the given customers failed. The bigger value of the penalty counter for a specified customer, the more difficult is to reinsert it into the solution [19].
After the random route is removed from the current solution, continuous at- tempts to include the ejected customers into the rest of the routes are performed (Alg. 3). These attempts are carried out until all customers from the Ejection Pool are inserted or the execution time of the algorithm reaches a specified time limit Tmax [2]. Sinsertfe (vi, σ) is defined as a set of the feasible partial solutions that are obtained by inserting the customer vi into all insertion positions in σ.
If Sinsertfe (vi, σ) is empty then the function Squeeze (line 7 in Alg. 3) is called in order to help the insertion of the selected customer into σ. The idea of this func- tion is to choose the temporarily infeasible insertion with the minimal penalty function value Fp(σ) defined in [19].
If the Squeeze function fails then it informs that the selected customer vi was not inserted into the solution σ. The value of the penalty counter for a chosen customer is then increased (line 10 in Alg. 3) and after that the ejections of the
1 begin
2 σ ← RemoveRouteInitStep(σ)
3 choose and remove a route randomly from the solutionσ
4 initializeEP with the customers from the removed route
5 initialize the penalty countersp[vi]← 1(i = 1, ..., N)
6 if Tcurr < Tmaxthen
7 whileEP= ∅ do
8 σ ← RouteEliminationStep(σ)
9 if cremodfc = 0 then
10 T C ← CooperationStep (σ)
11 if T C = true then
12 break
13 end
14 end
15 else
16 T C ← CooperationStep (σ)
17 end
18 if EP= ∅ then
19 restoreσ to the beginning solution
20 end
21 returnσ
22 end
Algorithm 2. RemoveRoute(σ) - minimizing the number of routes (cre - route elimination steps count; fc- frequency of the processes communication)
1 begin
2 cre← cre+ 1// increase route elimination steps count
3 select and eject the customer vi from EP using LIFO stack
4 if Sinsertfe (vi, σ) = ∅ then
5 σ ← the new solution σ selected randomly fromSinsertfe (vi)
6 else
7 σ ← Squeeze (vi,σ)
8 end
9 if vi is not inserted intoσ then
10 p[vi]← p[vi] + 1
11 selectσ fromSej(vi) with min.Psum=p[vout(1)] + ... +p[vout(km)]
12 updateσ ← σ
13 add the ejected customers:v(1)out,v(2)out, ...,vout(km) to the EP
14 σ ← Perturb (σ)
15 end
16 returnσ
17 end
Algorithm 3. RouteEliminationStep(σ) - single elimination step
customers are tested (up to the limit km [19]). For this reason, the set Sej(vi) is being constructed, that contains the solutions with various ejected customers and the customers inserted at different route positions. The Perturb function (line 14 in Alg. 3) is used to diversify the search and only the feasible partial solutions are accepted in that step [2].
The parameter fc indicates how often the processes communicate with each other to exchange the number of routes and the best solutions found to date. This parameter is compared with the route elimination steps count creof the current process (line 6 in Alg. 2). The chain of the route elimination steps of the process P2is updated to the better solution between the best solutions (in our algorithm only with respect to the number of routes) found by processes P1 and P2. Such a cooperation is continued until the process PN obtains the best solution from all the processes P1to PN−1and after that, the best result is sent back to the first process P1. This way of the cooperation between the processes guarantees that after each cooperation step the process P1holds the best solution found to date by all the processes used in the communication chain. It also guarantees that each process Pi holds the best solution found in the previous cooperation step by all the processes. The scheme of the cooperation of the processes is presented in Fig. 1.
In order to minimize the cost of the cooperation step during an algorithm execution, the nonblocking operations for both receive and send functions were introduced. Because of the asynchronous communication, each time at the end of a cooperation step the process has to wait for already started operations to be finished. The successive route elimination steps are performed during that step to take advantage of a waiting time in the beneficial way. A two-grading system of sending data from one process to another was introduced. In the first step only a number of routes is sent to the next process, along with the information whether a solution will be sent as well or not. The full solution is sent only if the number of routes of the current solution is lower than in last cooperation step.
σ1(init)⇒ σ2(init)⇒ σ3(init)⇒ • • • ⇒ σN−1(init)⇒ σN(init)⇒ σ(init)1
↓ ↓ ↓ ↓ ↓ ↓
σ1(1fc) ⇒ σ2(1fc) ⇒ σ(1f3 c) ⇒ • • • ⇒ σ(1fN−1c) ⇒ σ(1fN c) ⇒ σ1(1fc)
↓ ↓ ↓ ↓ ↓ ↓
σ1(2fc) ⇒ σ2(2fc) ⇒ σ(2f3 c) ⇒ • • • ⇒ σ(2fN−1c) ⇒ σ(2fN c) ⇒ σ1(2fc)
↓ ↓ ↓ ↓ ↓ ↓
• • • • • • • • •
↓ ↓ ↓ ↓ ↓ ↓
σ(xf1 c) ⇒ σ2(xfc) ⇒ σ3(xfc) ⇒ • • • ⇒ σN−1(xfc) ⇒ σ(xfN c) ⇒ σ(xf1 c)
Fig. 1. Scheme of cooperation of the processes (σi(init)- initial solution of the processPi; σi(jfc) - j cooperation step of the processPi; x - cooperation steps count)
1 begin
2 if processID = 0 then
3 T C ← (Tcurr > Tmax)
4 PerformSendStep (σ)
5 PerformReceiveStep (σ)
6 else
7 PerformReceiveStep (σ)
8 PerformSendStep (σ)
9 end
10 WaitForSentRequests (σ)
11 return T C
12 end
Algorithm 4. CooperationStep(σ) - function of cooperation of processes
3 The MPI Implementation
In the MPI implementation of the parallel algorithm all processes Pi, i = 1, 2, . . . , N execute the parallel loop shown in Alg. 1. The processes communicate repeat- edly with the same parameters list and for this reason the persistent communi- cation requests have been created for all send and receive operations:
– SendNoReq {a persistent request of sending number of routes}, – SendSolReq {a persistent request of sending full solution}, – RecvNoReq {a persistent request of receiving number of routes}, – RecvSolReq {a persistent request of receiving full solution}.
The list of the communication parameters is binded to the persistent requests only once using MPI Send init or MPI Recv init function (line 2 in Alg. 1).
When the cooperation between the processes is finished the persistent requests have to be freed (line 9 in Alg. 1) using MPI Request free function.
In the MPI version of the algorithm, MPI Start () function is used to initiate the communication with a persistent request handle and in the algorithm it is used for both send and receive operations.
Only after both send and receive operations were invoked, it is necessery to wait for the sent requests to be finished. By taking advantage of the MPI Test function it was possible to test if they are already completed. If not, the successive route elimination steps are executed.
4 Experimental Results
The MPI implementation was run on the Galera (TASK) supercomputer which nodes consisted of 2 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 s). The computer was executing Linux operating system. The source code was compiled using Intel 10.1 compiler and MVAPICH1 v. 0.9.9 MPI library.
The selected benchmarking tests proposed by Gehring and Homberger (GH) were used for testing the MPI implementation of the parallel algorithm. The investigations reported in [2] indicated that using the sequential version of the algorithm most of the GH tests can be solved quickly to good accuracy except of tests from C1, C2, RC1, RC2 groups having 400-1000 customers. In order to analyze speedup of the parallel version of this algorithm only those of the GH tests were chosen for which the execution time was one of the longest. The results of the experiments are gathered in Tab. 1 and illustrated in Fig. 2. Overall 150 experiments for RC tests (RC2 6 2, RC2 8 5, RC2 10 5) and 100 experiments for C tests (C2 6 2, C2 8 9, C2 10 7) for 1..16 and for 20, 24, 28, ..., 64 proces- sors were carried out. The results in the table show speedups and the average execution times only for 1, 16, 32, 48 and 64 processors. It can be noticed that the algorithm achieves slightly worse speedups for tests from C1/C2 groups than for tests from RC1/RC2 groups. It may indicate that the additional operations such as communication, idling and synchronization of processes take the larger part for the C1/C2 tests than for RC1/RC2 tests.
By running the parallel algorithm on 32/64 processors 12 new best-known solutions were found and they are reported in Tab. 2.
Table 1. Results for selected GH tests (p – number of processors, ¯τ – average execution time [s],S – speedup, exp. – number of experiments)
RC2 6 2 RC2 8 5 RC2 10 5
p τ¯ S exp. τ¯ S exp. τ¯ S exp.
1 2573.1 1.00 150 369.7 1.00 150 454.7 1.00 150 16 203.9 12.62 150 30.4 12.16 150 35.6 12.64 150 32 91.1 28.24 150 13.3 27.81 150 16.8 27.02 150 48 62.8 40.99 150 9.5 39.01 150 11.9 38.21 150 64 52.4 49.12 150 7.6 48.97 150 9.5 47.88 150
C2 6 2 C2 8 9 C2 10 7
p τ¯ S exp. τ¯ S exp. τ¯ S exp.
1 492.1 1.00 100 1564.6 1.00 100 852.7 1.00 100 16 49.2 10.01 100 152.5 10.27 100 90.1 9.47 100 32 19.8 24.92 100 64.6 24.21 100 40.6 21.02 100 48 14.3 34.48 100 49.2 31.82 100 26.1 32.70 100 64 11.0 44.66 100 36.5 42.92 100 21.4 39.92 100
1 16 32 48 64
1 16 32 48 64
S
p C2 6 2 Ideal
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p
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Fig. 2. Speedup S vs. number of processors p for tests C2 6 2, C2 8 9, C2 10 7, RC2 6 2, RC2 8 5 and RC2 10 5
Table 2. New best-known solutions (N – number of customers, p – number of processors)
Instances N p Best-known New-best Time [s]
C2 6 9 600 64 18 17 113
RC2 6 1 600 64 15 14 192
RC2 6 5 600 32 12 11 378
RC2 8 1 800 32 19 18 360
RC2 8 2 800 64 17 16 80
RC2 8 5 800 32 16 15 232
C1 10 2 1000 32 91 90 61
C2 10 3 1000 64 29 28 9980
C2 10 4 1000 32 29 28 463
C2 10 6 1000 32 30 29 394
C2 10 7 1000 32 30 29 3005
C2 10 10 1000 64 29 28 542
5 Conclusions
The parallel algorithm for minimizing the number of routes in the VRPTW along with its MPI implementation were presented. The main advantage of the proposed algorithm is a very short time of obtaining solutions which contain the number of routes equal to or better than the best known solutions. The parallel algorithm displays very good performance in terms of the quality of solutions, robustness and computational effort. The experimental results obtained for the selected tests by Gehring and Homberger indicated that the algorithm scales better for the tests from groups RC1/RC2 than for the tests from groups C1/C2.
It may be expected, that the proposed way of parallelization will lead to similar results also in case of other combinatorial optimization problems.
Acknowledgments. We thank the following computing centers where the com- putations of our project were carried out: Academic Computer 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 Modeling, Warsaw Univer- sity (computing grant G27-9), Wroclaw Centre for Networking and Supercom- puting (computing grant 30).
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