Local Search with Annealing-like Restarts to Solve ... - Semantic Scholar
Local Search with Annealing-like Restarts to Solve the Vehicle Routing Problem with Time Windows Haibing Li and Andrew Lira
Department of Computer Science National University of Singapore Singapore, 117543 {lihb, alim}acomp.nus.edu.sg
ABSTRACT In this paper, we propose a metaheuristic based on snneedinglike restarts to diversify and intensify local searches for solving the vehicle m u t i n g problem with time windows (VR_PTW). Using the Solomon's benchmark instances for the problem, our method obtained 7 new best results and equaled 19 other best results. Extensive comparisons indicate that our method is comparable to the best published literatures.
Keywords routing, vehicle routing problem with time windows, local search, diversification, intensification
1.
INTRODUCTION
propose a tabu-embedded simulated annealing metaheuristic with K-restart strategy to local search procedure. We apply these ideas to solve the V R P T W . The rest parts of the paper are orgemized as follows. In Section 2, the V R P T W is briefly described with reviewing of the related work. Section 3 describes the local search structures. I n Section 4, we present the metaheuristic in detei]s. Computational experiments are conducted in Section 5, in addition, results on 56 Solomon's b e n d t m a r k problem instances axe reported and our approach is compared with the best methods for this problem. Finally, we present our conclusions in Section 6.
2. NOTATION A N D T H E V R P T W 2.1 Notation
Local Search is commonly used to solve hard combinatorial optimization problems. A local search algorithm starts with an initial solution and searches for better solutions. T h e q,ml;ty of solutions obtained by loca~ search methods is directly influenced by definition of neighborhood space in the search process. Efficient generation of superior neighborhoods is important in order to have an eiTective search. Being trapped in local o p t i m a is the other factor affecting the q-Mity of solutions generated using local search. Advanced search strategies including simulated annealing, tabu search, and genetic algorithms were devised to overcome such problems. These search strategies are classified as metaheuristics, since they act as guiding strategies for their respective iocal search procedures.
n, m : number of customers, number of vehicle used, respectively. L: the m a x i m u m number of consecutively non-improving iterations in the tabu-embedded simulated annealing procedure. It is proposed to be a small number. K : the m a x i m u m number of restarts of the tabu-embedded simulated annealing procedure on the current best solution. C o s t ( S ) : objective function of solution H of local search.
SACost (8): objective function of solution S in tabu-embedded metropolis procedure in the simulated mmealing algorithm.
We proposed two ideas to improve local search effectiveness. In the aspect of neighborhood, we propose three constraintbased neighborhoods which enable local search to explore more promising solutions, while pruning infeasible and inferior solutions. In the aspect of local optima solutions, we
Dist(S), ST(S), W T ( S ) : travel distance, schedule time and waiting time of solution S, respectively. N=~ (S), N ~ (S), N, -+ (S): constraint-based neighborhoods of solution S obtained by shift operator, exchange operator and rearrange operator respectively.
*Dr Andrew Lira is the contact author Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage, and that copies bear this notice and the full citation on the first page. To copy otherwise, to relmblish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee.
N (S): N (S) : N ~ (S)UN¢~ (8) are the neighborhoods where local searches are conducted. ~b: the depth of K-ary tree due to K restarts. To, T : initial and global annealing t e m p e r a t u r e respectively.
SAC 2002, Madrid, Spain {~ 2002 A C M 1-58113-445-2/02/03/...$5.00
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'~, 0, 7, A: p e r ~ l t y factor for m~ Diat(S), ST(S), W T ( $ ) respectively.
reported in Saveishergh [1] [9] and 1~alllard e t a / . [11]. Russell [12] proposed a composite heuristic which mixes route construction and route improvement procedures.
Vpo,: the customer node at t h e position o f p o s in a route. Dp~,ta~o,a" the distance between customers at position p o s l and pose in a route.
Metaheuristice are a new generation of heuristics developed in the recent decade. Near o p t i m a l solutions for t h e V R P T W were produced b y t a b u search (l~c.hat & Semet [13], R o c h a t
2.2
~- a ~ a r d
The VRPTW
[14], Potvin e~ o~ [lS], ChiaaS & Rmsell [18],
~Paillard et aL [11], C o r d e a u et aft. [17]), simulated annealing (Chiang & Russell [18]), evolutionary algorithms (Thangiah [19], P o t v i n & Bengio [20], Homberger & Gehring [21]) and hybrid algorithms of the above three algorithms (Thanglah et aL [22] and Russell [12]). Surveys on the V R P T W are referred to Desrochers et al. [23] [24] and Cordeau et M. [25].
Let G = (V, A) be a digraph. V = VH U {vo} is the node set where VJv = { ~ E V i i = 1 , 2 , . . . , ~ } represents t h e customers and node vo denotes the d e p o t where a fleet of vehicles are housed. E a c h node v~ E V has an associat.ed customer d e m a n d qi(qo = 0), a service time si(so = 0) and a service-time window [e~, li]. For each pair of nodes (v~,vj')(i ~ j , i , j --- 9 , 1 , 2 , . . . , n ) , a non-negative distance d~j and a non-negative travel time t~i are known. Due to the time window constraints, arcs m a y n o t exist between some node pairs. Therefore, t h e arc set can be defined as A = {(,J,,vi)lv,,t,~ E V,t,, # ~,~,to, + s, + t,~ < l~.}. If a vehicle reaches a customer ~: before e~, it needs to wait until e¢ in order to service the customer. T h e schedule time of a route is the s u m of t h e waiting time, the service and travel time. T h e objective of the V I ~ T W is to service all customers w i t h o u t violating vehicle c a p a c i t y constraints and time window constraints with a m i n i m u m number of vehicles and, for the s a m e number of routes with the m i n i m u m travel distance, followed b y the m i n i m u m schedule time and the m i n i m u m waiting time.
3.
LOCAL SEARCH STRUCTURES
3.1
Neighborhood Generating Mechanisms
In our method, three edge-swapping operators are used to generate local search neighborhoods. These operators are, the shift operator, t h e exchange o p e r a t o r and the rearrange operator. Save]sbergh [1] [9] and Tall]ard et al. [11] also dealt with similar edge-exchange operators for the V R P T W . But t h e y restricted the length of edge segments to be swapped to certain m a x i m u m values. We do n o t impose such restrictions. Further, we use 3 simple conditions t h a t can be calculated in constant time to prune infeasible/inferior solutions, so t h a t the neighborhoods' sizes are reduced to accelerate the local searches. A m o n g these operators, the shift operator and the exchange o p e r a t o r are used for generating neighborhoods for local searches in the t a b u - e m b e d d e d simulated annealing procedure, while the rearrange o p e r a t o r is only used for post-processing on the local o p t i m a obtained
Saveisbergh [I] proved t h a t the V R P T W isA / P - h a r d . M u c h work has been done to derive exact algorithms to solve the problem. Desrochers st sl. [2] proposed a column generation a p p r o a c h t h a t solved seven of Solomon's b e n c h m a r k instances e x a c t l y p r o p o s e d in Solomon [3]. Fisher e t a / . [4] proposed a ir(-tree relaxation approach t h a t solved two of Solomon's b e n c h i n a r k instances. Other exact approaches can b e found in D u m a s et sl. [5], K o h l & M a d s e n [6] and Kolen et aL [7]. Because of the exponential time complexi t y of these approaches, it's unlikely t h a t these algorithms can produce o p t i m a l solutions for practical-sized V R P T W with reasonable c o m p u t a t i o n a l time. Hence, much work focused on generating near o p t i m a l solutions truing heuristic approaches.
by local searches.
3.1.1
The Shift Operator
Using the shift operator, none-null route segments (with at least one node) can be moved from one route to another route under the condition of not violating the cap a c i t y constraints and time window constraints. For each pair of selected routes, say, Route i and Route j (we assume 1 _< i < j _< m , i # j without loss of generality), the shii~ operator is used in ~.wo directions, so t h a t s sub-neighborhood NI=~j(S) can be o b t a i n e d by shifting route segments from Route i to Route j and another sub-neighborhood A r j ~ ( S ) can be obtained by shifting route segments from Route j to Route ~ We have denoted the neighborhood of solution S obtained b y the shift o p e r a t o r as N ~ (S), thus, N~, (S) ---
In general, heuristics for V R P T W can be classified as cLassical heuristics or metaheuristics. Classical heuristics include construction heuristics, improvement heuristics and composite heuristics. R o u t e construction heuristics build feasible solutions b y inserting a n u n r o u t e d customer into a current p a r t i a l route until all customers axe routed. T h e sequential insertion algorithm proposed by Solomon [3] belongs to construction heuristics. Its parallel version was implemented in P o t v i n & Rousseau [8]. These m e t h o d s often produce solutions very quickly, but the quality of these solutions m a y not be good. For this reason, construction hettristics are usually applied to o b t a i n initial solutions for improvement heuristics or other two-phase heuristics. R o u t e improvement heuristics modify a solution b y performing local searches for bet ter neighboring solutions within its neighborhoods generated b y n o d e / e d g e - s w a p p i n g operators. Edgeexchange heuristics are referred to Savelsbergh [1] [9] and P o t v i n & Rousseau [10]. Efficient implementations of edgeexchange for speeding up the pruning of inferior solutions are
U ? ~ ~ U?__,+AN,~j(s) o ~ r ~ , ( s ) } . Figure 1 shows an example o f shifting a route segment from Route 1 to Route 8. To accelerate the local searches, three simple conditions are considered to prune infeasible or inferior solutions in conatant time, during local searches within N=~(.S'). We use the example shown in Figure 1 to explain this. Firstly, if the t o t a l - c a p a c i t y of the route segment from poall to p o a l 2 to be shifted in Route I exceeds the left-capacity of Route 8, the solution obtained by shifting is infeasible so t h a t it can be discarded. We record the nsed-capacity-sofar in each route node so t h a t c o m p u t i n g t o t a l - c a p a c i t y and left-capacity can be done in c o n s t a n t time. Secondly, if either (~po,_p. . . . Vpo,tt) or (VpooXa,~po,) is an infeasible arc in
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exchange o p e r a t i o n can also be p r u n e d d u r i n g local searches using t h e s a m e three conditions. Of course, since m o r e arcs a n d r o u t e segments are involved in t h e exchange o p e r a t o r , the conditions axe a l i t t l e m o r e complex, However, all t h r e e conditions can also b e c a l c u l a t e d in c o n s t a n t time. A c t u ally, i t ' s possible to o b t a i n an exchange o p e r a t i o n by two shift operations. However, as we discussed previously, each shift o p e r a t i o n requires checking of 3 c o n d i t i o n s to p r u n e infeasible/Lr~erior solutions. Thus, two shift o p e r a t i o n s need to check 6 c o n d i q o n s . T h i s results in a d d i t i o n a l c o m p u t a t i o n a l t i m e and i m p l e m e n t a t i o n c o m p l e x i t y in c o n t r a s t to t h e exchange o p e r a t o r .
Here, (Ds~o.n_s,. . . . ~o,z~e®t "4- Dpo._p. . . . po, lZ + D ~ o , n . ~ o , ) is t h e t o t a l d i s t a n c e of t h e three newly c o n n e c t e d edges, while
(Dpom_~..,~oa~ + Dvo,~,~oon~.ffit + Dpo._~...ooo) is the t o t a l d i s t a n c e of t h e t h r e e old edges removed due to shifting. A < 0 means t h a t t h e travel d i s t a n c e w o n ' t be decreased, hence t h e s o l u t i o n o b t a i n e d b y the o p e r a t i o n does n o t result in a b e t t e r solution. However, as our first o b j e c t i v e is to minimize the n u m b e r of "vehicles, therefore, before excluding a shii~ o p e r a t i o n u n d e r t h e c o n d i t i o n of A < 0, we m u s t m a k e sure t h a t if this shift o p e r a t i o n will move t h e whole r o u t e s e g m e n t o u t of a r o u t e so t h a t we can elirrdnate one vehicle. Therefore, we will o n l y diseaxd a solution i f A ~_ 0 a n d t h e shifting o p e r a t i o n w o n ' t e l i m i n a t e a vehicle. As d e t e r m i n e d b y E q u a t i o n 1, this check c a n be performed in c o n s t a n t time_ Using these t h r e e conditions, ]N=~(S)[ can be reduced efficiently. We na~ne t h e solution space N ~ (~) c o n s t r a i n e d b y t h e three conditions as a c o n s t r a i n t - b a s e d n~dghbo~hood.
3.1.2
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F i g u r e 2: T h e E x c h a n g e
d i g r a p h G defined in S e c t i o n 2, t h e shif~ o p e r a t i o n ~ resuits in an infeasible solution t h a t can be discarded. Since the arc set A is initialized w h e n t h e g r a p h d a t a is generated, checking for this c o n d i t i o n can be done in c o n s t a n t t~me. Finally, since we axe only i n t e r e s t e d in seaxching for b e t t e r solutions, if an edge-shift incurs an i n c r e m e n t in the t o t a l t r a v e l d i s t a n c e of t h e two routes, t h e n this shift operation results in an inferior solution t h a t c a n be d i s c a r d e d too. To simplify t h e checking p r o c e d u r e such t h a t if an ine r e m e n t in t r a v e l d i s t a n c e incurs, we can use t h e following condition, let:
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3.1,3
The Rearrange Operator
Using t h e r e a r r a n ~ o p e r a t o r , r o u t e segments c a n be repositioned w i t h i n the s a m e r o u t e u n d e r t h e c o n d i t i o n of n o t v i o l a t i n g the t i m e window c o n s t r a i n t s . We have d e n o t e d t h e n e i g h b o r h o o d of solution B o b t a i n e d b y the r e a r r a n g e o p e r a t o r M N, -~ (3). F i g u r e 3 shows an exatnple of repositioning a r o u t e s e g m e n t to a n o t h e r p o s i t i o n w i t h i n a route. Sjm;larly, we can also get a c o n s t r a i n t - b a s e d n e i g h b o r h o o d N. -~ (S) using the second c o n d i t i o n a n d t h e t h i r d c o n d i t i o n discussed above. T h e first c o n d i t i o n is not needed because t h e c a p a c i t y rerr~ins u n c h a n g e d w h e n t h e r e a r r a n ~ n g ope r a t i o n is applied w i t h i n one r o u t e .
The Exchange Operator
Using t h e exchange o p e r a t o r , non-null r o u t e segments can be exchanged b e t w e e n two routes u n d e r t h e c o n d i t i o n of not violating c a p a c i t y a n d time window constraints. We have d e n o t e d t h e n e i g h b o r h o o d of solution S o b t a i n e d by this o p e r a t o r as N¢~ (S). F i g u r e 2 shows an e x a m p l e of exchanging two r o u t e segments b e t w e e n Rou~e I to Roule ~. Like t h e shift o p e r a t o r , infeasible or inferior solutions o b t a i n e d by
3.1.4 562
Objectives a n d C o s t F u n c t i o n in L o c a l Search
The order of the objectives to be minimized for the V R P T W is: 1) the number of vehicles; 2) the traveling cost; 3) the schedule time; 4) the drivers' total waiting time to begin service. To reflect this order, we adopt the following cost function:
Oost(S) ----¢,m + ~Diat(S) +-/ST(S) + A W T ( S )
(2)
T h e penalty weight factors are set to be: a ~ / 9 ~> 7 ~:~ A. Of course, we can change the setting of these factors to get different orders of the four objectives.
O Neighbor obtained by T M P • Neighbor obtained by T M P * T D L $
4. T H E M E T A H E U R I S T I C 4.1 The Algorithm
F i g u r e 4: D i v e r s i f i c a t i o n b y TSA
The algorithm is proposed as a two-phase framework: 1.3 I F A < 0 T H E N p r o b 4 - 1 P h a s e 1: Obtain an initial solution for the V R P T W a heuristic.
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1.4 E L S E prob +- e-AT 1.5 I F random(0,1) _< prob T H E N
Phase 2: Start local search from the initial solution, using tabu-embedded simulated aanealing ( T S A ) with K restart strategy.
1.5.1 Accept S" 1.5.2 U p d a t e annealing temperature: T +-- aT 1.5.3 Record S' into t a b u list 2. R E T U R N
In phase 1, initial solutions are obtained by the insertion heuristic proposed by Solomon [3]. These initial solutions are then used to feed the local searches in phase 2, which we will discuss in details as follows. Firstly, the pseudocode of T S A is described as follows:
In the procedure, SACost(s) is a cost function to evaluate solutions on behalf of simulated annealing. It is defined to adjust the cooling procedure with setting of To, T and 6. It can be defined as Equation 3.
TSA(Solution S)
ALGORITHM
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S ACost (S) = Dist( S)÷~ox W T ( S), where ~ = 0.01 x D i s t ( S )
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1. Sb e- S; N o n l m p r o v i n g +- 0
Finally, the algorithm which combines T e A with the bestbased K - r e s t a r t strategy is as follows:
2. W H I L E N o n l m p r o v i n g < L D O 2.1 Use T M P ( S ) to obtain a non-tabu S' E N ( S )
I N P U T : an initial solution So O U T P U T : a best solution So found so far A L G O R I T H M K - T S A ( S o l u t l o n So)
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3. ]~LETURN Sb
K c , t h e t r a v e l d i s t a n c e t e n d s t o c h a n g e b e t w e e n a cons t a n t , while t h e t e n d e n c y of c o m p u t a t i o n a J t i m e c h a n g e s n o t so a b r u p t l y as d o e s for K < K c , if w e a n a l y z e t h e m b y reg r e s s i o n m o d e l s . T h e m a i n r e a s o n m a y lie i n t h a t , a K > /tf~ i n t r o d u c e s ( K -- K s ) r e s t a r t s for a d d i t i o n a l c o m p u t a t i o n a l t i m e , b u t h a s l i t t l e influence o n c o n v e r g e n c e b e h a v i v o r o f s o l u t i o n s a f t e r K~.
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l a r g e r than ~ we g e t T(frL,~) ----O(~). T h e r e f o r e , t h e t i m e c o m p l e x i t y o f K - T S A is O(~K-~).
S.
Comparison with O t h e r Heuristics
T a b l e 2 c o m p a r e s t h e m e a n n u m b e r of vehicles ( M N V ) a n d t h e m e a n t r a v e l d i s t a n c e (_~TD) o b t a i n e d b y o u r a l g o r i t h m s a n d o t h e r h e u r i s t i c s for t h e V ~ L P T W . T h i s t a b l e shows t h a t , o u r r e s u l t s m a t c h t h e b e s t r e s u l t s o n d a t a s e t s C 1 artd C 2 . W e get b e s t _~NV o n R 1 w i t h C L M . O n R 2 a n d R C 1 , o u r r e s u l t s a r e m o d e r a t e . Finally~ o n R C 2 , we o b t a i n e d b e s t ]~4NV w i t h C R a n d H G . w h i l e t h e ]kJ'TD o n t h i s d a t a s e t is o n l y s l i g h t l y infer£or t o t h a t of I-1(3. T h e c o m p a r i s o n shows t h a t o u r a p p r o a c h is c o m p e t i t i v e w i t h t h o s e h e u r i s t i c s o n s o l v i n g t h e V R P T W . F r o m q ~ b l e 2, o u r r e s u l t s m a t c h t h e b e s t r e s u l t s on d a t a sets C 1 a n d C 2 . W e g o t b e s t M N V o n R 1 w i t h C L M . O n R2 a n d R C 1 , o u r r e s u l t s a r e m o d e r a t e . F i n a l l y , o n R C 2 , we o b t a i n e d b e s t M N V w i t h C R , H G a n d t h e M T D o n t h i s d a t a set is o n l y s l i g h t l y inferior to t h a t o f HG.
I n ~swh i t e r a t i o n , t h e t i m e c o m p l e x i t y is m a i n l y d e t e r m i n e d b y l o c a l s e a r c h e s i n t h e n e i g h b o r h o o d s t h a t we defined. L e t T ( m , n) b e t h e o v e r a l l t i m e c o m p l e x i t y of l o c a l s e a r c h e s in the constraint-based neighborhoods. T h e n the time complexity of K - T S A is O(~K) w T(m~ n), w h e r e T(m, n) is a n a l y z e d as follows:
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E x p e r i m e n t s on Solomon's Problems
W e c o n d u c t e d c o m p u t a t i o n s / e x p e r i m e n t s o n S o l o m o n ' s 55 b e n c h m a r k p r o b l e m s ( S o l o m o n [3]). T h e h a r d w a r e u s e d to r u n o u r e x p e r i m e n t s is t h e P C P e n t i u m I I I 54fiMHZ. T h e ~ l g o r i t h m was d e v e l o p e d in t h e C-l-+ progr~n_m£ng l a n g u a g e with the Standard Template L i b r a r y (STL). T h e O S u s e d is t h e l i n u x k e r n e l 2.4.0. T h e v a l u e s of t h e p a r a m e t e r s u s e d are To = 50, 6 ----0.95, K ----2 0 [ T ] , L = 4.
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JiG 10.00 828.38 3.00 589.86 11.92 1220.97 ~.73
968.55 11.50
1s88.~ 3.~5
~.4o.4s
ULM
HFb'KP
~ur8
10.0(3
10.00
828.38 3.00 589.86
828.38 3.00
10.00 828.38
1£,08
2.73 969.58 II.50 1389.78 3.25 1134.52
589.86 12.41 1200.54 3 936.509 12 1383.21 3.38 1116.51
3.00
589.86 12.08 1215_14 2.91 953.428 11.75
1385.47 3_25 1142.48
Table 2: C o m p a r i s o n s o f B e s t A v e r a g e s o n all D a t a Sets
[4] M. Fisher, K. J~cnsten, and O. M~dsen. Vehicle routing with time windows: T w o optimization algorithx~. OpemHons Re#earch, 45:488-492, 1997.
855 85o
i 845 840
[5] Y. Dumas, J. Desrosiers, E. Gelinea, and M. Solomon. An optimal algorithm for the travelin K salesman problem with time windows. Operat~or~ Reaearch, 43:367-371, 1995.
-- 835 | 83O
~- 825 820
[6] N. Kohl and O- Madsen. A n optimization algorithm for the vehicle routing problem with time windows based on laKrangian relaxation. Opera/Wr~ Research,
300
45:395-406, 1997. 150
~
[7] A. Kolen, A. Rinnooy, and H. Trienehens. Vehicle
routing with time windows. Ope~tior~ Research, 35:266--273, 1987.
so
[8] J. Potvin and J. Rousseau. A parallelroute building algorithms for the vehicle routing and scheduling problem with time windows. F~ropean 3ourna/of Opem~iona/Research, 66:331-340, 1993.
|
0
20
~
40
60
F i g u r e 6: Sermitivity o f K b y E x p e r i m e n t a t i o n
6.
[9] M. Save]sbergh. The vehicle routing problem with time windows: Minimizing route duration. ORSA
CONCLUSION
Journal on Computing, 4:146-154, 1992_
In this paper, we proposed a tabu-embedded simulated annealing (TSA) metaheuristic which runs with best-ba~ed Krestarts strategy, to diversify and intensify local searches for solving combinatorial optimization problpxn~ as a general framework. The diversification by T S A and the intensification b y / ( - r e s t a r t strategy was discussed with details. The sensitivity of K is analyzed by experimentation. The computational results show that our algorithm is comparable to other approaches for solving the VRPTW. The approach can be extended to solve other variants of VRP (Li & Lira [28]) and other combinatorial optimization problems.
7.
[10] J. Potvin and J. Rousseau. An exchange heuristic for routing problems with time windows. Journal of
Operational Research Society, 46:1433-1446, 1995. [11] E. Taillard, P. Badeau, M. Gendreau, F. Geurtin, and J. Potvin. A tabu search heuristic for the vehicle routing problem with time windows. Transpor~a~on Science, 31:170-186, 1997. [12] R. Russell. Hybrid heuristics for the vehicle routing problem with time windows. ~anaporCation Science, 29:156-166, 1995.
REFERENCES
[13] Y. Rochat and F. Semet. A tabu search approach for
[1] M. Saveisbergh. Local search for routing problems
de~vering pet food and flour in switzerland. Journal
with time windows. Annals of Operatio~ Research,
of Operational Research SocietF, 45:1233-1246, 1994.
4:285-305, 1985.
[14] Y. Rochat and E. Taillard. Probabilistic diversification and intensification in local search for vehicle routing.
[2] M. Desrochers, J. Desrosiers, and M. Solomon. A new optimization algorithm for the vehicle routing problem with time windows. Operations Research, 40:342--354, 1992.
Journal of Heuristics, 1:147-167, 1995. [15] J. Potvin, B. Garcia, and L. Rousseau. The vehicle routing problem with time windows - part 1: 13~bu search. INFORMS Journal on Computing, 8:158-184, 1996.
[3] M. Solomon. Algorithms for the vehicle routing and scheduling problems with time window constrains. Operations Research, 35:254-264, 1987.
565
/'rob 1%103
R104
1%107
Ft204
NG 5 12 7 8 6 6 10 7 6 7 8 8 10 12 11 11 13 10 10 11 10 12 10 II 11 8 11 8 12 g 10 10 50
50
1~207
49
51
RC106
FLC202
9 8 9 10 9 Ii 8 10 10 7 9 32 31 37
Table
De~a£1a of R o ~ ~ e . a 0-40-53-12-68--80-0 0-92-96-14-44-38-86-16-61-85-91-100-37-0 0-36-64-49-63-90-32-70-0 0-71-65-78-34-35-81-77-26-0 0-50-33-76-79-10-31-0 0-94-96-95-97-87-13-0 0- 7-19-11-8-46-47-48-82-18-89-0 0-52-62-88,-84-17-93--59-0 0-60-45-83-5-99-6-0 0-2-22-75-56-4-25-54-0 0-26-39-23-67-55-24-29-3-0 0-27-69-30-9-66-20-51-1-0 0-42-43-15-57-41-74-72-73-21-58-0 92-98-I 4-44--39-86-16-61-85-91-100-37 42-43--15-87-57-41-2 2-74-73--21-26 52-88-62-11-64--63-90-32--10-31-27 6-94-96-59-93-99-84-17-45-83-5-60-89 18-82-48-46-8-47-36-49-19-7 70-30-20-66-9-35-65-71-51-50 28-1-69-76- 5 3-40-2-13-95-97-58 72-75-5 6-23- 67-39-55-4-25- 54 68-80-24-29-79-78-34-81-33-3-77-12 0-26-21-39-23-67-55-24-29-3--77-0 0-27-69-1-76-79-78-34-68-80-12-28-0 0--52-7-62-11-64-49-19--48-82-18--89-0 0-40-53-6-96-59-95-13-58-0 0-60-83--45-5-99--87-57-2-74-72-73-0 0-20-65-9-66-71-35-81-50-0 0-94-92-98-44-14-38-86-16-91-100-37-97-0 0-47-36-46-8-84-17-61-85-93-0 0--42-43-15-41-22-75-56-4-25-54-0 0-33-51-30-88-31-10-63-90-32-70-0 0-27-52-18-7-82-48--19-11-62-88--31-69-76-12-67-39-53-29-79-8 I-9-51-20-66-65-71-35 -34-78-29-24-55-4-72-74-73-21-2-13-97-37 -98-100-91-85-93-59-96-6-89-0 0-94-95-92-42-57-22-75-56-23-41-15-43-14 -44-38--86-16-61-99--87-60-83-8-84-5-17-45 -46-47- 36-49-64-63-90-32-10-30- 70-1-5 033-3-77-68-80-54-25-25-40-58--0 0-1-50-33--81-65-34-29-24--39--67-23-15--4314-44-38-86-16-84-5-99-6-18--8-48--47-4919-10-63-90-32-66-71-35-20-70-317-82-17 -61-85-91-100-37-98-93-96-0 0-42-92-59-60-83-45-46-36-64-11-62-86-5227-69-30-51-9-78-79-3-76-28-53-40-2-8757-41-22-73-21-72-74-75-56-4-25-55-54-80 -68-77-12-26-58-13--97-95-94-89-0 0-69-98-88-53-12-10-9-13-17-0 0 - 1 4 - 1 1 - 8 7- 5 9- 7 5 - 9 7 - 5 8 - - 7 4 - 0 0-95-62-63-85-76-51-84-56--66-0 0-42-44-39-40-36-38-41-43--37-35-0 0-72--71-67-30-32-34-50-93-80-0 0-2-45-5-8,-7-6--4(~ A 3-1-100-0 0-92-61-81-90-94-96-54-68--0 0-82-52-99-86-57-22-49-20-24-91-0 0-65-83-64-19-23-21-18-48-25-77-0 0-33-31-29-27-28-26-89-0 0-15-16-47-78-73-79-60-55-70-0 0-65-82-12-14-47-15-11-69-64-19-23-48.-1676-51-22-86-87-9-57-52-10-97-59-74-13-177-4-(]0-100-70-0 0-98-45-5-3-1-42-36-37-39-44-61-88-73-1699-53-76-79-6-6-46-2-55-68-43-35-54-9693-94-80-0 0-92-95-85-63-33-28-26-27-29--31-30-62-6771-72-38-40-41-81-90-91-84-49-20-83-6656-50-34-32- 89- 24-21-25- 77-75-58-0 S- New
Best
Solutions 566
Obtained
by
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Teat Best Published Casea N V/TD Re]. C101* -10/828.94 RT C102" 10/828.94 RT C103" 10/828.06 FIT C104" 10/824.78 RT C105" 10/828.94 11T CI06" 10/828.94 R T C108" 10/828.94 R T C109" 10/828.94 RT C201" '3/591.56 RT C202" 3/591.56 RT C203" 3/591.17 RT C204" 3/590.60 RT C205" 3/588.88 RT C206" 3/588.49 C207" 3/588.29 RT C208" 3/588.32 RT R101* ' 19/1650.80 RT R102 17/1486.12 117 R103"* 13/1292.85 HG R104"* 9/1013.32 HG 11105 14/1377.11 11T R106 12/1252.03 RT R107** 10/1113.69 C L M RI08 9/964.38 CLM R109 11/1194-73 HG Rl10 10/1124.40 RGP Rlll 10/1099.46 HG RII2 9/1003.73 HG R201" 411252.37 H G R202 311191.70 R G P R203 31942.70 HG R204"* 2 1849.62 CLM R205 3 f994.42 RGP R206 3/912.97 RT R207"* 2/914.39 RT R208 2/730.771 BFSKP 11209 3/909.86 RGP R210 3/939.373 BFSKP R211 2/910.09 HG RCI01 14/1696.94 T B G G P RCI02 12/1554.75 TBGGP RC103 11/1262.02 RT RC104 10/1135.48 C L M RC105 13/1833.72 R G P RC108"* 11/1427.13 C L M RC107 11/1230.54 T B G G P RCI08 10/1139.82, T B G G P 11C201 4/1046.94 CLM RC202"* 3/1389.57 HG RC203 3/1060.45 HG RC204 3/799.12 HG RC205 4/1302.42 HG RC206 3/1153.93 RGP RC207 3/1062.05 CLM RC208 3/829.69 RGP
[16] W. Chiang and R. Russell.A reactivetabu search metaheuristic for the vehiclerouting problem with time windows. INFORMS Jou~al on Oomputir.d, 9:417-430, 1097.
Our Beat Resul~ NV/TD CT 10/8~8.g4 140
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[17] J. Cordeau, G. Laporte, and A. Mercier. A ,-i6ed tabu search heuristic for vehicle routing problems with time windows. Technica/Report CRT-00-03, Center for Research on Transportation, Montreal, C~,-,-Ja, 2000.
16z
[18} W. Chiang and R. Russell. Simulated anneA]i,g metaheuristics for the vehicle routing problem with time windows. Annals of Operations Research, 63:3--27, 1996. [19] S. Thangiah. Vehicle routing with time windows using genetic algorithms. Technical Report SRU-CPSC-TR-93-23, Computer Science Department, SZippery Reck University, Slippery Rock, PA, 1993.
1084 s/sas.4g 1124 S/588.£9 1159 8/588.8~ 1249 19/1650.80 835 17/1486.41 1078 13/1~9~.68 1926 9/I00~31 2200 14/1381.37 975 12/1269.72 2296
lo/1~o4.~ 9/986.25
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15/1658.62 13/1513.60 11/1319-99
1356 1344 2016 1492 3325 3355 1510 4637 4470 3196 4856 6500 2805 3565 4481 1006 1631 505
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[20] J. Potvin and S. Bengio. The vehicle routing problem with time windows - part 2: Genetic search. I N F O R M S Journal on Computing, 8:165-172, 1996. [21] J. Homberger and H. Gehring. Two evolutionary metsheuristics for the vehicle routing problem with time windows. INFOR, 37:297-318, 1999.
1263
[22] S. Tha~giah, I. Osman, and T. Sun. Hybrid genetic algorithm, simulated annealing and tabu search methods for vehicle routing problems with time windows. " I ~ n i c a l Report UKC/OR94/4, Institute of Mathematics and Statistics, University of Kent, Canterbury, UK, 1994.
[23] J. Desrochers, J. Leustra, M. Savelsbergh, and F. Sourais. Vehicle routing with time windows: Optimization mad approximation. In Vehicle Routing: Methods and Studies. Elsevier Science Publishers, North-Holland, 1988. [24] J. Desrosiers, Y. Dumas, M. Solomon, and F. Sorrels. Time constrained routing and scheduling. In Handbook~ in Operations Reaegrch and Management Science: Network Routing. Elsevier Science Publishers, Amsterdam, 1995. [25] J. Cozdeau, G. Desanlniers, J. Deerosiers, M. Solomon, and P. Soumls. The vrp with time windows. In The Vehicle Routing Problem. SIAM, Philadelphia, PA, Forthcoming. [26] B. Backer, V. Fttrnon, P. Shaw, P. Kilby, and P. Prosser. Solving vehicle routing problem using constraint prograntrning and metahearistics. Journal o] Heuristics, 6:501-523, 2000.
2039
[27] L. Rousseau, M. Gendreau, and G. Pesant. Using constraint-based operators to solve the vehicle routing problem with time windows. Journal of Heuriatics, Forthcoming.
2475
:2398
[28] H. Li and A. Lisa. A metaheurlstic for the pickup and delivery problem with time windows. In ICTAI BOOI, November 2001.
Table 1: Our Solutions va Best Published Solutions
567
Department of Computer Science National University of Singapore Singapore, 117543 {lihb, alim}acomp.nus.edu.sg
ABSTRACT In this paper, we propose a metaheuristic based on snneedinglike restarts to diversify and intensify local searches for solving the vehicle m u t i n g problem with time windows (VR_PTW). Using the Solomon's benchmark instances for the problem, our method obtained 7 new best results and equaled 19 other best results. Extensive comparisons indicate that our method is comparable to the best published literatures.
Keywords routing, vehicle routing problem with time windows, local search, diversification, intensification
1.
INTRODUCTION
propose a tabu-embedded simulated annealing metaheuristic with K-restart strategy to local search procedure. We apply these ideas to solve the V R P T W . The rest parts of the paper are orgemized as follows. In Section 2, the V R P T W is briefly described with reviewing of the related work. Section 3 describes the local search structures. I n Section 4, we present the metaheuristic in detei]s. Computational experiments are conducted in Section 5, in addition, results on 56 Solomon's b e n d t m a r k problem instances axe reported and our approach is compared with the best methods for this problem. Finally, we present our conclusions in Section 6.
2. NOTATION A N D T H E V R P T W 2.1 Notation
Local Search is commonly used to solve hard combinatorial optimization problems. A local search algorithm starts with an initial solution and searches for better solutions. T h e q,ml;ty of solutions obtained by loca~ search methods is directly influenced by definition of neighborhood space in the search process. Efficient generation of superior neighborhoods is important in order to have an eiTective search. Being trapped in local o p t i m a is the other factor affecting the q-Mity of solutions generated using local search. Advanced search strategies including simulated annealing, tabu search, and genetic algorithms were devised to overcome such problems. These search strategies are classified as metaheuristics, since they act as guiding strategies for their respective iocal search procedures.
n, m : number of customers, number of vehicle used, respectively. L: the m a x i m u m number of consecutively non-improving iterations in the tabu-embedded simulated annealing procedure. It is proposed to be a small number. K : the m a x i m u m number of restarts of the tabu-embedded simulated annealing procedure on the current best solution. C o s t ( S ) : objective function of solution H of local search.
SACost (8): objective function of solution S in tabu-embedded metropolis procedure in the simulated mmealing algorithm.
We proposed two ideas to improve local search effectiveness. In the aspect of neighborhood, we propose three constraintbased neighborhoods which enable local search to explore more promising solutions, while pruning infeasible and inferior solutions. In the aspect of local optima solutions, we
Dist(S), ST(S), W T ( S ) : travel distance, schedule time and waiting time of solution S, respectively. N=~ (S), N ~ (S), N, -+ (S): constraint-based neighborhoods of solution S obtained by shift operator, exchange operator and rearrange operator respectively.
*Dr Andrew Lira is the contact author Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage, and that copies bear this notice and the full citation on the first page. To copy otherwise, to relmblish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee.
N (S): N (S) : N ~ (S)UN¢~ (8) are the neighborhoods where local searches are conducted. ~b: the depth of K-ary tree due to K restarts. To, T : initial and global annealing t e m p e r a t u r e respectively.
SAC 2002, Madrid, Spain {~ 2002 A C M 1-58113-445-2/02/03/...$5.00
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560
'~, 0, 7, A: p e r ~ l t y factor for m~ Diat(S), ST(S), W T ( $ ) respectively.
reported in Saveishergh [1] [9] and 1~alllard e t a / . [11]. Russell [12] proposed a composite heuristic which mixes route construction and route improvement procedures.
Vpo,: the customer node at t h e position o f p o s in a route. Dp~,ta~o,a" the distance between customers at position p o s l and pose in a route.
Metaheuristice are a new generation of heuristics developed in the recent decade. Near o p t i m a l solutions for t h e V R P T W were produced b y t a b u search (l~c.hat & Semet [13], R o c h a t
2.2
~- a ~ a r d
The VRPTW
[14], Potvin e~ o~ [lS], ChiaaS & Rmsell [18],
~Paillard et aL [11], C o r d e a u et aft. [17]), simulated annealing (Chiang & Russell [18]), evolutionary algorithms (Thangiah [19], P o t v i n & Bengio [20], Homberger & Gehring [21]) and hybrid algorithms of the above three algorithms (Thanglah et aL [22] and Russell [12]). Surveys on the V R P T W are referred to Desrochers et al. [23] [24] and Cordeau et M. [25].
Let G = (V, A) be a digraph. V = VH U {vo} is the node set where VJv = { ~ E V i i = 1 , 2 , . . . , ~ } represents t h e customers and node vo denotes the d e p o t where a fleet of vehicles are housed. E a c h node v~ E V has an associat.ed customer d e m a n d qi(qo = 0), a service time si(so = 0) and a service-time window [e~, li]. For each pair of nodes (v~,vj')(i ~ j , i , j --- 9 , 1 , 2 , . . . , n ) , a non-negative distance d~j and a non-negative travel time t~i are known. Due to the time window constraints, arcs m a y n o t exist between some node pairs. Therefore, t h e arc set can be defined as A = {(,J,,vi)lv,,t,~ E V,t,, # ~,~,to, + s, + t,~ < l~.}. If a vehicle reaches a customer ~: before e~, it needs to wait until e¢ in order to service the customer. T h e schedule time of a route is the s u m of t h e waiting time, the service and travel time. T h e objective of the V I ~ T W is to service all customers w i t h o u t violating vehicle c a p a c i t y constraints and time window constraints with a m i n i m u m number of vehicles and, for the s a m e number of routes with the m i n i m u m travel distance, followed b y the m i n i m u m schedule time and the m i n i m u m waiting time.
3.
LOCAL SEARCH STRUCTURES
3.1
Neighborhood Generating Mechanisms
In our method, three edge-swapping operators are used to generate local search neighborhoods. These operators are, the shift operator, t h e exchange o p e r a t o r and the rearrange operator. Save]sbergh [1] [9] and Tall]ard et al. [11] also dealt with similar edge-exchange operators for the V R P T W . But t h e y restricted the length of edge segments to be swapped to certain m a x i m u m values. We do n o t impose such restrictions. Further, we use 3 simple conditions t h a t can be calculated in constant time to prune infeasible/inferior solutions, so t h a t the neighborhoods' sizes are reduced to accelerate the local searches. A m o n g these operators, the shift operator and the exchange o p e r a t o r are used for generating neighborhoods for local searches in the t a b u - e m b e d d e d simulated annealing procedure, while the rearrange o p e r a t o r is only used for post-processing on the local o p t i m a obtained
Saveisbergh [I] proved t h a t the V R P T W isA / P - h a r d . M u c h work has been done to derive exact algorithms to solve the problem. Desrochers st sl. [2] proposed a column generation a p p r o a c h t h a t solved seven of Solomon's b e n c h m a r k instances e x a c t l y p r o p o s e d in Solomon [3]. Fisher e t a / . [4] proposed a ir(-tree relaxation approach t h a t solved two of Solomon's b e n c h i n a r k instances. Other exact approaches can b e found in D u m a s et sl. [5], K o h l & M a d s e n [6] and Kolen et aL [7]. Because of the exponential time complexi t y of these approaches, it's unlikely t h a t these algorithms can produce o p t i m a l solutions for practical-sized V R P T W with reasonable c o m p u t a t i o n a l time. Hence, much work focused on generating near o p t i m a l solutions truing heuristic approaches.
by local searches.
3.1.1
The Shift Operator
Using the shift operator, none-null route segments (with at least one node) can be moved from one route to another route under the condition of not violating the cap a c i t y constraints and time window constraints. For each pair of selected routes, say, Route i and Route j (we assume 1 _< i < j _< m , i # j without loss of generality), the shii~ operator is used in ~.wo directions, so t h a t s sub-neighborhood NI=~j(S) can be o b t a i n e d by shifting route segments from Route i to Route j and another sub-neighborhood A r j ~ ( S ) can be obtained by shifting route segments from Route j to Route ~ We have denoted the neighborhood of solution S obtained b y the shift o p e r a t o r as N ~ (S), thus, N~, (S) ---
In general, heuristics for V R P T W can be classified as cLassical heuristics or metaheuristics. Classical heuristics include construction heuristics, improvement heuristics and composite heuristics. R o u t e construction heuristics build feasible solutions b y inserting a n u n r o u t e d customer into a current p a r t i a l route until all customers axe routed. T h e sequential insertion algorithm proposed by Solomon [3] belongs to construction heuristics. Its parallel version was implemented in P o t v i n & Rousseau [8]. These m e t h o d s often produce solutions very quickly, but the quality of these solutions m a y not be good. For this reason, construction hettristics are usually applied to o b t a i n initial solutions for improvement heuristics or other two-phase heuristics. R o u t e improvement heuristics modify a solution b y performing local searches for bet ter neighboring solutions within its neighborhoods generated b y n o d e / e d g e - s w a p p i n g operators. Edgeexchange heuristics are referred to Savelsbergh [1] [9] and P o t v i n & Rousseau [10]. Efficient implementations of edgeexchange for speeding up the pruning of inferior solutions are
U ? ~ ~ U?__,+AN,~j(s) o ~ r ~ , ( s ) } . Figure 1 shows an example o f shifting a route segment from Route 1 to Route 8. To accelerate the local searches, three simple conditions are considered to prune infeasible or inferior solutions in conatant time, during local searches within N=~(.S'). We use the example shown in Figure 1 to explain this. Firstly, if the t o t a l - c a p a c i t y of the route segment from poall to p o a l 2 to be shifted in Route I exceeds the left-capacity of Route 8, the solution obtained by shifting is infeasible so t h a t it can be discarded. We record the nsed-capacity-sofar in each route node so t h a t c o m p u t i n g t o t a l - c a p a c i t y and left-capacity can be done in c o n s t a n t time. Secondly, if either (~po,_p. . . . Vpo,tt) or (VpooXa,~po,) is an infeasible arc in
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m
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•
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po,,t2 posL2 ~ n p o m ~ Customer Node
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- - - -4i
po,
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Ds, o..~.... ,o.)
O)
exchange o p e r a t i o n can also be p r u n e d d u r i n g local searches using t h e s a m e three conditions. Of course, since m o r e arcs a n d r o u t e segments are involved in t h e exchange o p e r a t o r , the conditions axe a l i t t l e m o r e complex, However, all t h r e e conditions can also b e c a l c u l a t e d in c o n s t a n t time. A c t u ally, i t ' s possible to o b t a i n an exchange o p e r a t i o n by two shift operations. However, as we discussed previously, each shift o p e r a t i o n requires checking of 3 c o n d i t i o n s to p r u n e infeasible/Lr~erior solutions. Thus, two shift o p e r a t i o n s need to check 6 c o n d i q o n s . T h i s results in a d d i t i o n a l c o m p u t a t i o n a l t i m e and i m p l e m e n t a t i o n c o m p l e x i t y in c o n t r a s t to t h e exchange o p e r a t o r .
Here, (Ds~o.n_s,. . . . ~o,z~e®t "4- Dpo._p. . . . po, lZ + D ~ o , n . ~ o , ) is t h e t o t a l d i s t a n c e of t h e three newly c o n n e c t e d edges, while
(Dpom_~..,~oa~ + Dvo,~,~oon~.ffit + Dpo._~...ooo) is the t o t a l d i s t a n c e of t h e t h r e e old edges removed due to shifting. A < 0 means t h a t t h e travel d i s t a n c e w o n ' t be decreased, hence t h e s o l u t i o n o b t a i n e d b y the o p e r a t i o n does n o t result in a b e t t e r solution. However, as our first o b j e c t i v e is to minimize the n u m b e r of "vehicles, therefore, before excluding a shii~ o p e r a t i o n u n d e r t h e c o n d i t i o n of A < 0, we m u s t m a k e sure t h a t if this shift o p e r a t i o n will move t h e whole r o u t e s e g m e n t o u t of a r o u t e so t h a t we can elirrdnate one vehicle. Therefore, we will o n l y diseaxd a solution i f A ~_ 0 a n d t h e shifting o p e r a t i o n w o n ' t e l i m i n a t e a vehicle. As d e t e r m i n e d b y E q u a t i o n 1, this check c a n be performed in c o n s t a n t time_ Using these t h r e e conditions, ]N=~(S)[ can be reduced efficiently. We na~ne t h e solution space N ~ (~) c o n s t r a i n e d b y t h e three conditions as a c o n s t r a i n t - b a s e d n~dghbo~hood.
3.1.2
.
F i g u r e 2: T h e E x c h a n g e
d i g r a p h G defined in S e c t i o n 2, t h e shif~ o p e r a t i o n ~ resuits in an infeasible solution t h a t can be discarded. Since the arc set A is initialized w h e n t h e g r a p h d a t a is generated, checking for this c o n d i t i o n can be done in c o n s t a n t t~me. Finally, since we axe only i n t e r e s t e d in seaxching for b e t t e r solutions, if an edge-shift incurs an i n c r e m e n t in the t o t a l t r a v e l d i s t a n c e of t h e two routes, t h e n this shift operation results in an inferior solution t h a t c a n be d i s c a r d e d too. To simplify t h e checking p r o c e d u r e such t h a t if an ine r e m e n t in t r a v e l d i s t a n c e incurs, we can use t h e following condition, let:
+
.
.~0" . . . . . -@.~
,,,
Depot
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A
.
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.
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Route 1 I - - - - - . , O Rente
R o u t e :2
-m
posl2_ne~
po,d2
......
3.1,3
The Rearrange Operator
Using t h e r e a r r a n ~ o p e r a t o r , r o u t e segments c a n be repositioned w i t h i n the s a m e r o u t e u n d e r t h e c o n d i t i o n of n o t v i o l a t i n g the t i m e window c o n s t r a i n t s . We have d e n o t e d t h e n e i g h b o r h o o d of solution B o b t a i n e d b y the r e a r r a n g e o p e r a t o r M N, -~ (3). F i g u r e 3 shows an exatnple of repositioning a r o u t e s e g m e n t to a n o t h e r p o s i t i o n w i t h i n a route. Sjm;larly, we can also get a c o n s t r a i n t - b a s e d n e i g h b o r h o o d N. -~ (S) using the second c o n d i t i o n a n d t h e t h i r d c o n d i t i o n discussed above. T h e first c o n d i t i o n is not needed because t h e c a p a c i t y rerr~ins u n c h a n g e d w h e n t h e r e a r r a n ~ n g ope r a t i o n is applied w i t h i n one r o u t e .
The Exchange Operator
Using t h e exchange o p e r a t o r , non-null r o u t e segments can be exchanged b e t w e e n two routes u n d e r t h e c o n d i t i o n of not violating c a p a c i t y a n d time window constraints. We have d e n o t e d t h e n e i g h b o r h o o d of solution S o b t a i n e d by this o p e r a t o r as N¢~ (S). F i g u r e 2 shows an e x a m p l e of exchanging two r o u t e segments b e t w e e n Rou~e I to Roule ~. Like t h e shift o p e r a t o r , infeasible or inferior solutions o b t a i n e d by
3.1.4 562
Objectives a n d C o s t F u n c t i o n in L o c a l Search
The order of the objectives to be minimized for the V R P T W is: 1) the number of vehicles; 2) the traveling cost; 3) the schedule time; 4) the drivers' total waiting time to begin service. To reflect this order, we adopt the following cost function:
Oost(S) ----¢,m + ~Diat(S) +-/ST(S) + A W T ( S )
(2)
T h e penalty weight factors are set to be: a ~ / 9 ~> 7 ~:~ A. Of course, we can change the setting of these factors to get different orders of the four objectives.
O Neighbor obtained by T M P • Neighbor obtained by T M P * T D L $
4. T H E M E T A H E U R I S T I C 4.1 The Algorithm
F i g u r e 4: D i v e r s i f i c a t i o n b y TSA
The algorithm is proposed as a two-phase framework: 1.3 I F A < 0 T H E N p r o b 4 - 1 P h a s e 1: Obtain an initial solution for the V R P T W a heuristic.
using
1.4 E L S E prob +- e-AT 1.5 I F random(0,1) _< prob T H E N
Phase 2: Start local search from the initial solution, using tabu-embedded simulated aanealing ( T S A ) with K restart strategy.
1.5.1 Accept S" 1.5.2 U p d a t e annealing temperature: T +-- aT 1.5.3 Record S' into t a b u list 2. R E T U R N
In phase 1, initial solutions are obtained by the insertion heuristic proposed by Solomon [3]. These initial solutions are then used to feed the local searches in phase 2, which we will discuss in details as follows. Firstly, the pseudocode of T S A is described as follows:
In the procedure, SACost(s) is a cost function to evaluate solutions on behalf of simulated annealing. It is defined to adjust the cooling procedure with setting of To, T and 6. It can be defined as Equation 3.
TSA(Solution S)
ALGORITHM
S'
S ACost (S) = Dist( S)÷~ox W T ( S), where ~ = 0.01 x D i s t ( S )
(8)
1. Sb e- S; N o n l m p r o v i n g +- 0
Finally, the algorithm which combines T e A with the bestbased K - r e s t a r t strategy is as follows:
2. W H I L E N o n l m p r o v i n g < L D O 2.1 Use T M P ( S ) to obtain a non-tabu S' E N ( S )
I N P U T : an initial solution So O U T P U T : a best solution So found so far A L G O R I T H M K - T S A ( S o l u t l o n So)
e
2.2 S b +-- Perform T D L S within N ( S ' ) , - P e r f o r m T D L S within N . - ,
2.4 I F Cost(S'b) < Cost(Sb) T H E N 1- Configure paxarneters: K , L, 6 mad To; T ~-- To
Sb 4- S'b; Nonlmprouing e- 0 2.5 E L S E Nonlmpro~ing 4- N o n l m p r o v i n 9 + I
2- Set t a b u list to be empty
2.6 S 4- S'b
3. Sg 4- So; gNonlmproulng 4- 0
4. W H I L E g N o n l m p r o v i n g < K D O
3. ]~LETURN Sb
K c , t h e t r a v e l d i s t a n c e t e n d s t o c h a n g e b e t w e e n a cons t a n t , while t h e t e n d e n c y of c o m p u t a t i o n a J t i m e c h a n g e s n o t so a b r u p t l y as d o e s for K < K c , if w e a n a l y z e t h e m b y reg r e s s i o n m o d e l s . T h e m a i n r e a s o n m a y lie i n t h a t , a K > /tf~ i n t r o d u c e s ( K -- K s ) r e s t a r t s for a d d i t i o n a l c o m p u t a t i o n a l t i m e , b u t h a s l i t t l e influence o n c o n v e r g e n c e b e h a v i v o r o f s o l u t i o n s a f t e r K~.
m will never
l a r g e r than ~ we g e t T(frL,~) ----O(~). T h e r e f o r e , t h e t i m e c o m p l e x i t y o f K - T S A is O(~K-~).
S.
Comparison with O t h e r Heuristics
T a b l e 2 c o m p a r e s t h e m e a n n u m b e r of vehicles ( M N V ) a n d t h e m e a n t r a v e l d i s t a n c e (_~TD) o b t a i n e d b y o u r a l g o r i t h m s a n d o t h e r h e u r i s t i c s for t h e V ~ L P T W . T h i s t a b l e shows t h a t , o u r r e s u l t s m a t c h t h e b e s t r e s u l t s o n d a t a s e t s C 1 artd C 2 . W e get b e s t _~NV o n R 1 w i t h C L M . O n R 2 a n d R C 1 , o u r r e s u l t s a r e m o d e r a t e . Finally~ o n R C 2 , we o b t a i n e d b e s t ]~4NV w i t h C R a n d H G . w h i l e t h e ]kJ'TD o n t h i s d a t a s e t is o n l y s l i g h t l y infer£or t o t h a t of I-1(3. T h e c o m p a r i s o n shows t h a t o u r a p p r o a c h is c o m p e t i t i v e w i t h t h o s e h e u r i s t i c s o n s o l v i n g t h e V R P T W . F r o m q ~ b l e 2, o u r r e s u l t s m a t c h t h e b e s t r e s u l t s on d a t a sets C 1 a n d C 2 . W e g o t b e s t M N V o n R 1 w i t h C L M . O n R2 a n d R C 1 , o u r r e s u l t s a r e m o d e r a t e . F i n a l l y , o n R C 2 , we o b t a i n e d b e s t M N V w i t h C R , H G a n d t h e M T D o n t h i s d a t a set is o n l y s l i g h t l y inferior to t h a t o f HG.
I n ~swh i t e r a t i o n , t h e t i m e c o m p l e x i t y is m a i n l y d e t e r m i n e d b y l o c a l s e a r c h e s i n t h e n e i g h b o r h o o d s t h a t we defined. L e t T ( m , n) b e t h e o v e r a l l t i m e c o m p l e x i t y of l o c a l s e a r c h e s in the constraint-based neighborhoods. T h e n the time complexity of K - T S A is O(~K) w T(m~ n), w h e r e T(m, n) is a n a l y z e d as follows:
be
E x p e r i m e n t s on Solomon's Problems
W e c o n d u c t e d c o m p u t a t i o n s / e x p e r i m e n t s o n S o l o m o n ' s 55 b e n c h m a r k p r o b l e m s ( S o l o m o n [3]). T h e h a r d w a r e u s e d to r u n o u r e x p e r i m e n t s is t h e P C P e n t i u m I I I 54fiMHZ. T h e ~ l g o r i t h m was d e v e l o p e d in t h e C-l-+ progr~n_m£ng l a n g u a g e with the Standard Template L i b r a r y (STL). T h e O S u s e d is t h e l i n u x k e r n e l 2.4.0. T h e v a l u e s of t h e p a r a m e t e r s u s e d are To = 50, 6 ----0.95, K ----2 0 [ T ] , L = 4.
COMPUATIONAL EXPERIMENTS 564
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C1 MTD G2 MNV 02 MTD R1 MNV R1 MTD 1£2 MNV R2 MTD B.C1 MNV RC1 MTD RC2 MNV RC2 MTD
8P,8.88 3.00
589. 86 12.25
1208.50 2.91 961,72 11.88 1377.39 3.38 1119.59
U/l I0.00 828.38 3.00 591.42 12.17 1204.19 2.73 986.32 II.875 1397.44 3.25 1229.54
TBGGP 10.00 828.38
3.00
589.86 12.17 1209.35 2.82 980.27
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1389.22 3.38
1117.44
JiG 10.00 828.38 3.00 589.86 11.92 1220.97 ~.73
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589.86 12.41 1200.54 3 936.509 12 1383.21 3.38 1116.51
3.00
589.86 12.08 1215_14 2.91 953.428 11.75
1385.47 3_25 1142.48
Table 2: C o m p a r i s o n s o f B e s t A v e r a g e s o n all D a t a Sets
[4] M. Fisher, K. J~cnsten, and O. M~dsen. Vehicle routing with time windows: T w o optimization algorithx~. OpemHons Re#earch, 45:488-492, 1997.
855 85o
i 845 840
[5] Y. Dumas, J. Desrosiers, E. Gelinea, and M. Solomon. An optimal algorithm for the travelin K salesman problem with time windows. Operat~or~ Reaearch, 43:367-371, 1995.
-- 835 | 83O
~- 825 820
[6] N. Kohl and O- Madsen. A n optimization algorithm for the vehicle routing problem with time windows based on laKrangian relaxation. Opera/Wr~ Research,
300
45:395-406, 1997. 150
~
[7] A. Kolen, A. Rinnooy, and H. Trienehens. Vehicle
routing with time windows. Ope~tior~ Research, 35:266--273, 1987.
so
[8] J. Potvin and J. Rousseau. A parallelroute building algorithms for the vehicle routing and scheduling problem with time windows. F~ropean 3ourna/of Opem~iona/Research, 66:331-340, 1993.
|
0
20
~
40
60
F i g u r e 6: Sermitivity o f K b y E x p e r i m e n t a t i o n
6.
[9] M. Save]sbergh. The vehicle routing problem with time windows: Minimizing route duration. ORSA
CONCLUSION
Journal on Computing, 4:146-154, 1992_
In this paper, we proposed a tabu-embedded simulated annealing (TSA) metaheuristic which runs with best-ba~ed Krestarts strategy, to diversify and intensify local searches for solving combinatorial optimization problpxn~ as a general framework. The diversification by T S A and the intensification b y / ( - r e s t a r t strategy was discussed with details. The sensitivity of K is analyzed by experimentation. The computational results show that our algorithm is comparable to other approaches for solving the VRPTW. The approach can be extended to solve other variants of VRP (Li & Lira [28]) and other combinatorial optimization problems.
7.
[10] J. Potvin and J. Rousseau. An exchange heuristic for routing problems with time windows. Journal of
Operational Research Society, 46:1433-1446, 1995. [11] E. Taillard, P. Badeau, M. Gendreau, F. Geurtin, and J. Potvin. A tabu search heuristic for the vehicle routing problem with time windows. Transpor~a~on Science, 31:170-186, 1997. [12] R. Russell. Hybrid heuristics for the vehicle routing problem with time windows. ~anaporCation Science, 29:156-166, 1995.
REFERENCES
[13] Y. Rochat and F. Semet. A tabu search approach for
[1] M. Saveisbergh. Local search for routing problems
de~vering pet food and flour in switzerland. Journal
with time windows. Annals of Operatio~ Research,
of Operational Research SocietF, 45:1233-1246, 1994.
4:285-305, 1985.
[14] Y. Rochat and E. Taillard. Probabilistic diversification and intensification in local search for vehicle routing.
[2] M. Desrochers, J. Desrosiers, and M. Solomon. A new optimization algorithm for the vehicle routing problem with time windows. Operations Research, 40:342--354, 1992.
Journal of Heuristics, 1:147-167, 1995. [15] J. Potvin, B. Garcia, and L. Rousseau. The vehicle routing problem with time windows - part 1: 13~bu search. INFORMS Journal on Computing, 8:158-184, 1996.
[3] M. Solomon. Algorithms for the vehicle routing and scheduling problems with time window constrains. Operations Research, 35:254-264, 1987.
565
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9 8 9 10 9 Ii 8 10 10 7 9 32 31 37
Table
De~a£1a of R o ~ ~ e . a 0-40-53-12-68--80-0 0-92-96-14-44-38-86-16-61-85-91-100-37-0 0-36-64-49-63-90-32-70-0 0-71-65-78-34-35-81-77-26-0 0-50-33-76-79-10-31-0 0-94-96-95-97-87-13-0 0- 7-19-11-8-46-47-48-82-18-89-0 0-52-62-88,-84-17-93--59-0 0-60-45-83-5-99-6-0 0-2-22-75-56-4-25-54-0 0-26-39-23-67-55-24-29-3-0 0-27-69-30-9-66-20-51-1-0 0-42-43-15-57-41-74-72-73-21-58-0 92-98-I 4-44--39-86-16-61-85-91-100-37 42-43--15-87-57-41-2 2-74-73--21-26 52-88-62-11-64--63-90-32--10-31-27 6-94-96-59-93-99-84-17-45-83-5-60-89 18-82-48-46-8-47-36-49-19-7 70-30-20-66-9-35-65-71-51-50 28-1-69-76- 5 3-40-2-13-95-97-58 72-75-5 6-23- 67-39-55-4-25- 54 68-80-24-29-79-78-34-81-33-3-77-12 0-26-21-39-23-67-55-24-29-3--77-0 0-27-69-1-76-79-78-34-68-80-12-28-0 0--52-7-62-11-64-49-19--48-82-18--89-0 0-40-53-6-96-59-95-13-58-0 0-60-83--45-5-99--87-57-2-74-72-73-0 0-20-65-9-66-71-35-81-50-0 0-94-92-98-44-14-38-86-16-91-100-37-97-0 0-47-36-46-8-84-17-61-85-93-0 0--42-43-15-41-22-75-56-4-25-54-0 0-33-51-30-88-31-10-63-90-32-70-0 0-27-52-18-7-82-48--19-11-62-88--31-69-76-12-67-39-53-29-79-8 I-9-51-20-66-65-71-35 -34-78-29-24-55-4-72-74-73-21-2-13-97-37 -98-100-91-85-93-59-96-6-89-0 0-94-95-92-42-57-22-75-56-23-41-15-43-14 -44-38--86-16-61-99--87-60-83-8-84-5-17-45 -46-47- 36-49-64-63-90-32-10-30- 70-1-5 033-3-77-68-80-54-25-25-40-58--0 0-1-50-33--81-65-34-29-24--39--67-23-15--4314-44-38-86-16-84-5-99-6-18--8-48--47-4919-10-63-90-32-66-71-35-20-70-317-82-17 -61-85-91-100-37-98-93-96-0 0-42-92-59-60-83-45-46-36-64-11-62-86-5227-69-30-51-9-78-79-3-76-28-53-40-2-8757-41-22-73-21-72-74-75-56-4-25-55-54-80 -68-77-12-26-58-13--97-95-94-89-0 0-69-98-88-53-12-10-9-13-17-0 0 - 1 4 - 1 1 - 8 7- 5 9- 7 5 - 9 7 - 5 8 - - 7 4 - 0 0-95-62-63-85-76-51-84-56--66-0 0-42-44-39-40-36-38-41-43--37-35-0 0-72--71-67-30-32-34-50-93-80-0 0-2-45-5-8,-7-6--4(~ A 3-1-100-0 0-92-61-81-90-94-96-54-68--0 0-82-52-99-86-57-22-49-20-24-91-0 0-65-83-64-19-23-21-18-48-25-77-0 0-33-31-29-27-28-26-89-0 0-15-16-47-78-73-79-60-55-70-0 0-65-82-12-14-47-15-11-69-64-19-23-48.-1676-51-22-86-87-9-57-52-10-97-59-74-13-177-4-(]0-100-70-0 0-98-45-5-3-1-42-36-37-39-44-61-88-73-1699-53-76-79-6-6-46-2-55-68-43-35-54-9693-94-80-0 0-92-95-85-63-33-28-26-27-29--31-30-62-6771-72-38-40-41-81-90-91-84-49-20-83-6656-50-34-32- 89- 24-21-25- 77-75-58-0 S- New
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Solutions 566
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11
Approach
Teat Best Published Casea N V/TD Re]. C101* -10/828.94 RT C102" 10/828.94 RT C103" 10/828.06 FIT C104" 10/824.78 RT C105" 10/828.94 11T CI06" 10/828.94 R T C108" 10/828.94 R T C109" 10/828.94 RT C201" '3/591.56 RT C202" 3/591.56 RT C203" 3/591.17 RT C204" 3/590.60 RT C205" 3/588.88 RT C206" 3/588.49 C207" 3/588.29 RT C208" 3/588.32 RT R101* ' 19/1650.80 RT R102 17/1486.12 117 R103"* 13/1292.85 HG R104"* 9/1013.32 HG 11105 14/1377.11 11T R106 12/1252.03 RT R107** 10/1113.69 C L M RI08 9/964.38 CLM R109 11/1194-73 HG Rl10 10/1124.40 RGP Rlll 10/1099.46 HG RII2 9/1003.73 HG R201" 411252.37 H G R202 311191.70 R G P R203 31942.70 HG R204"* 2 1849.62 CLM R205 3 f994.42 RGP R206 3/912.97 RT R207"* 2/914.39 RT R208 2/730.771 BFSKP 11209 3/909.86 RGP R210 3/939.373 BFSKP R211 2/910.09 HG RCI01 14/1696.94 T B G G P RCI02 12/1554.75 TBGGP RC103 11/1262.02 RT RC104 10/1135.48 C L M RC105 13/1833.72 R G P RC108"* 11/1427.13 C L M RC107 11/1230.54 T B G G P RCI08 10/1139.82, T B G G P 11C201 4/1046.94 CLM RC202"* 3/1389.57 HG RC203 3/1060.45 HG RC204 3/799.12 HG RC205 4/1302.42 HG RC206 3/1153.93 RGP RC207 3/1062.05 CLM RC208 3/829.69 RGP
[16] W. Chiang and R. Russell.A reactivetabu search metaheuristic for the vehiclerouting problem with time windows. INFORMS Jou~al on Oomputir.d, 9:417-430, 1097.
Our Beat Resul~ NV/TD CT 10/8~8.g4 140
10/828.9~ lO/868.06
164 18o
10/8P~.78 I0/8~8.9~
412 170
1o/s~8.g4 I0/828.g4 ~o/8~8.g4
203 175
3/591.56 3/591.56 3/591.17
1356 1102 1334
s/5g0.60
1348
R/58& 88
[17] J. Cordeau, G. Laporte, and A. Mercier. A ,-i6ed tabu search heuristic for vehicle routing problems with time windows. Technica/Report CRT-00-03, Center for Research on Transportation, Montreal, C~,-,-Ja, 2000.
16z
[18} W. Chiang and R. Russell. Simulated anneA]i,g metaheuristics for the vehicle routing problem with time windows. Annals of Operations Research, 63:3--27, 1996. [19] S. Thangiah. Vehicle routing with time windows using genetic algorithms. Technical Report SRU-CPSC-TR-93-23, Computer Science Department, SZippery Reck University, Slippery Rock, PA, 1993.
1084 s/sas.4g 1124 S/588.£9 1159 8/588.8~ 1249 19/1650.80 835 17/1486.41 1078 13/1~9~.68 1926 9/I00~31 2200 14/1381.37 975 12/1269.72 2296
lo/1~o4.~ 9/986.25
908
11/1208.96 10/1159.35 11/1066.32 10/967.88 4/1252.37 4/1084.767 3/949.396
15/1658.62 13/1513.60 11/1319-99
1356 1344 2016 1492 3325 3355 1510 4637 4470 3196 4856 6500 2805 3565 4481 1006 1631 505
lO/1141.o9
683
13/1637.62 H/Ij~4.Ts 11/1240.66 10/1147.42 4/1425.21
748 1201 282 1278
s/zsT#.n7
3578
3/1088.53
3/818.66
1956 3186
4/1304.64
1753
3/1159.o3 3/1107.16
3970
~/8J~05 3/1032.55 3/931.62
~/90~13 2/732.8 3/950.59 3/1018.95
3/8o1.81
3/862.34
[20] J. Potvin and S. Bengio. The vehicle routing problem with time windows - part 2: Genetic search. I N F O R M S Journal on Computing, 8:165-172, 1996. [21] J. Homberger and H. Gehring. Two evolutionary metsheuristics for the vehicle routing problem with time windows. INFOR, 37:297-318, 1999.
1263
[22] S. Tha~giah, I. Osman, and T. Sun. Hybrid genetic algorithm, simulated annealing and tabu search methods for vehicle routing problems with time windows. " I ~ n i c a l Report UKC/OR94/4, Institute of Mathematics and Statistics, University of Kent, Canterbury, UK, 1994.
[23] J. Desrochers, J. Leustra, M. Savelsbergh, and F. Sourais. Vehicle routing with time windows: Optimization mad approximation. In Vehicle Routing: Methods and Studies. Elsevier Science Publishers, North-Holland, 1988. [24] J. Desrosiers, Y. Dumas, M. Solomon, and F. Sorrels. Time constrained routing and scheduling. In Handbook~ in Operations Reaegrch and Management Science: Network Routing. Elsevier Science Publishers, Amsterdam, 1995. [25] J. Cozdeau, G. Desanlniers, J. Deerosiers, M. Solomon, and P. Soumls. The vrp with time windows. In The Vehicle Routing Problem. SIAM, Philadelphia, PA, Forthcoming. [26] B. Backer, V. Fttrnon, P. Shaw, P. Kilby, and P. Prosser. Solving vehicle routing problem using constraint prograntrning and metahearistics. Journal o] Heuristics, 6:501-523, 2000.
2039
[27] L. Rousseau, M. Gendreau, and G. Pesant. Using constraint-based operators to solve the vehicle routing problem with time windows. Journal of Heuriatics, Forthcoming.
2475
:2398
[28] H. Li and A. Lisa. A metaheurlstic for the pickup and delivery problem with time windows. In ICTAI BOOI, November 2001.
Table 1: Our Solutions va Best Published Solutions
567
