a new tabu search heuristic for the open vehicle routing problem

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A New Tabu Search Heuristic for the Open Vehicle Routing Problem Author(s): Z. Fu, R. Eglese and L. Y. O. Li Source: The Journal of the Operational Research Society, Vol. 56, No. 3 (Mar., 2005), pp. 267- 274 Published by: Palgrave Macmillan Journals on behalf of the Operational Research Society Stable URL: http://www.jstor.org/stable/4102125 . Accessed: 03/12/2014 18:01 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . Palgrave Macmillan Journals and Operational Research Society are collaborating with JSTOR to digitize, preserve and extend access to The Journal of the Operational Research Society. http://www.jstor.org This content downloaded from 155.97.178.73 on Wed, 3 Dec 2014 18:01:57 PM All use subject to JSTOR Terms and Conditions

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Page 1: A New Tabu Search Heuristic for the Open Vehicle Routing Problem

A New Tabu Search Heuristic for the Open Vehicle Routing ProblemAuthor(s): Z. Fu, R. Eglese and L. Y. O. LiSource: The Journal of the Operational Research Society, Vol. 56, No. 3 (Mar., 2005), pp. 267-274Published by: Palgrave Macmillan Journals on behalf of the Operational Research SocietyStable URL: http://www.jstor.org/stable/4102125 .

Accessed: 03/12/2014 18:01

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

Palgrave Macmillan Journals and Operational Research Society are collaborating with JSTOR to digitize,preserve and extend access to The Journal of the Operational Research Society.

http://www.jstor.org

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Page 2: A New Tabu Search Heuristic for the Open Vehicle Routing Problem

Journal of the Operational Research Society (2005) 56, 267-274 C 2005 Operational Research Society Ltd. All rights reserved. 0160-5682/05 $30.00

www.palgrave-journals.com/jors

A new tabu search heuristic for the open vehicle routing problem Z Fu'*, R Eglese2 and LYO Li3 'Central South University, Changsha, PR China; 2Lancaster University, Lancaster, UK; and 3The Hong Kong Polytechnic University, Hong Kong, PR China

In this paper, another version of the vehicle routing problem (VRP)-the open vehicle routing problem (OVRP) is studied, in which the vehicles are not required to return to the depot, but if they do, it must be by revisiting the customers assigned to them in the reverse order. By exploiting the special structure of this type of problem, we present a new tabu search heuristic for finding the routes that minimize two objectives while satisfying three constraints. The computational results are provided and compared with two other methods in the literature. Journal of the Operational Research Society (2005) 56, 267-274. doi: 10. 1057/palgrave.jors.2601817 Published online 25 August 2004

Keywords: vehicle routing; open vehicle routing problem; tabu search; distribution management

Introduction

The open vehicle routing problem (OVRP) is similar to the standard vehicle routing problem (VRP). The important feature of this problem, which distinguishes it from the basic VRP, is that the vehicles are not required to return to the depot, or if they are required to do so, it must be accomplished by revisiting the customers assigned to them in the reverse order. Therefore, the vehicle routes are not closed paths but open ones.

The basic OVRP can be described as follows. There is a depot and a set of customers with specified demands. At the depot is located a vehicle fleet. Every vehicle has its own specified capacity and operating cost. The travelling cost between the depot and the customers, as well as between any pair of customers is known. Then the problem consists of finding the set of routes that minimize two objectives: (1) the total number of vehicles required to serve all customers, and (2) total travelling cost; while satisfying the following three main constraints: (1) each route originates at the depot and terminates at one of the customers, (2) each customer is visited once and only once by exactly one vehicle and its demand is totally satisfied, and (3) the customers who are visited in each route have total demand less than or equal to the capacity of the vehicle assigned to serve the route.

It is assumed that the capital cost of an additional vehicle will always exceed any travelling costs that could be saved by its use. Therefore, the priority is given to the first objective.

Like the basic VRP, variants arise when the constraint of total length of a route is introduced, or the available vehicles are not all identical, and so on.

OVRPs are faced by a company that either does not own a vehicle fleet at all, or its vehicle fleet is inappropriate or inadequate to satisfy the demand of its customers. Therefore, the company has to contract all or part of its goods delivery or pickup to external carriers. The hired vehicles are not obliged to return to the depot and the cost to the company may depend on the distance travelled while loaded, that is, the length of the open routes. Other areas where applications can be found for the OVRP are rail, bus and air transportation, in which vehicles (trains, buses and aero- planes) usually perform open routes, and the problems are usually ones with simultaneous pickup and delivery, and time windows.

The problem can be divided into the following three types:

(1) Delivery only. The vehicles are assigned to delivery routes in which they do not have to return to the company's distribution centre (depot).

(2) Pickup only. The vehicles are assigned to pick-up routes in which they can start directly from the customers at the other end of the routes to the depot.

(3) Both delivery and pickup. After finishing all deliveries, vehicles return to the depot by following the delivery routes, visiting customers in the reverse order and picking up goods which must be sent to the distribution centre, or after finishing all pickups, back from the depot by following the pickup routes, delivering goods to customers in the reverse order. This is, in fact, a special case of the VRP with Backhauls, in which vehicles may

*Correspondence: Z Fu, School of Traffic and Transportation Engineer- ing, Central South University at Railway Campus, Changsha 410075, PR China. E-mail: [email protected]

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Page 3: A New Tabu Search Heuristic for the Open Vehicle Routing Problem

268 Journal of the Operational Research Society Vol. 56, No. 3

visit different customers by following different routes in the return trips.

Some other examples of the OVRP in the real world may be found in Schrage,1 as well as in Sariklis and Powell.2

In this paper, we present a tabu search heuristic for the OVRP with both vehicle capacity and route length constraints. Experimental results that are compared with other approaches are reported.

Literature review and problem analysis

The practical importance of the OVRP was established two decades ago,1 but only in recent years has it received some attention from theoretical researchers. Sariklis and Powell2 present a sequential two-phase heuristic method, clustering followed by routing, for the OVRP. In the routing phase of the method, the minimum spanning tree problem (MSTP) is solved for every formed cluster. If the solution is a chain then it is the optimal solution. If not, then both the second and the third stages of the routing phase must be performed, with a penalty procedure being employed. Brandaio3 proposed a tabu search (TS) algorithm for the OVRP with both vehicle capacity and route length constraints.

Furthermore, two case-oriented papers in the literature can be included in the category of the OVRP (even though the authors did not refer to the OVRP), and support the argument of the practical importance of the OVRP.

The first one was about a train plan model for British Rail freight services through the Channel Tunnel by Fu and

Wright.4 British Rail planned to operate direct high-speed freight train services between Britain and the Continent when the Channel Tunnel opened in 1994. Some wagons were assigned to trains with a single origin and a single destination. However, all other wagons from Britain to the Continent were picked up by trains from the marshalling depot in London, where new trains were formed that delivered the wagons to the stations on the Continent; while the wagons from the Continent to Britain were picked up by trains, in reverse order, to the marshalling depot in London. Here new trains were formed that delivered the wagons to the stations in Britain. Train planning in this context involved the specification of the origin and destination of

trains, together with moving routes, details of picking up or

delivering activities along the routes, to minimize the number of trains through the tunnel, the train mileage and the amount of marshalling, and to satisfy the train capacity. Time windows relative to the station opening times and time slots through the tunnel were also considered.

The second case study was due to Li and Fu,5 in which the school bus routing problem in Hong Kong was studied. The schools may run the service themselves, or just coordinate the service and contract out to individual drivers. In both cases, in the morning a bus leaves its parking place (usually near the driver's home, not the school), arrives at the first

pickup point where the route starts, and travels along a predetermined route, picking up the pupils at pickup points and taking them to school. In the afternoon, the procedure is reversed: a bus picks up pupils at their school and drops them off at their pickup points along the route in the reverse order, and return to its overnight parking place. As the starting and ending point of a school bus route is different, the route is an open one. The problem was to find the set of school bus routes that satisfies the bus capacity, to minimize the total number of buses required and the total travel time, etc.

In the two case study problems, an important common feature, which is different from the basic VRP and OVRP, is that each customer can be visited more than once by different vehicles, that is, the demand can be split and served by different vehicles. This feature gives more choice for constructing the routes, resulting in better utilization of vehicle capacity and fewer vehicles being required.

The elimination from the basic VRP of the constraint that vehicles must return to the depot does not actually result in an easier and less complex problem. In the OVRP each route is a Hamiltonian path, instead of a Hamiltonian cycle as happens in the VRP. As stated by Syslo et al,6 the minimum Hamiltonian path problem has been shown to be NP-hard. Therefore, the whole OVRP is also NP-hard, and so it is

appropriate to solve large-scale OVRPs by a heuristic method. Even though there are many good heuristics available for

the VRP, we will not attempt to convert the OVRP into a VRP as part of the solution process since good OVRP solutions can be very different from corresponding VRP solutions. For, as stated by Syslo et al,6 'deleting the largest edge from a minimum Hamiltonian cycle does not

necessarily yield the minimum Hamiltonian path in the network.'

Notation

Let G=(V, E) be a given undirected network, where

V= {0,... , n} is the vertex set and E is the edge set. Vertices

i= 1,...,n correspond to the customers, whereas vertex 0

corresponds to the depot. A solution of the problem can be denoted by a permutation (0, il, i2, i3, 0, i4, ...,0, in-1, i,,) of

(0,..., n). Only 0 is allowed to appear more than once, each time for a new route. The first 0 and the following vertices before the second 0 consist of the first delivery route, the second 0 and the following vertices before the third 0 consist of the second delivery route, and so on. For the pickup only problem, for convenience, the notation of a solution may be changed simply by putting the 0 at the end of a route, that is,

(il, i2, i3, 0, i4,...,0, i,,1, inO). In consideration of feasibility, for the delivery problem the first item of a solution must be 0, while for the pickup problem the last item must be 0. If one 0 is adjacent to another, that means one route contains no customer, that is, can be eliminated.

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Page 4: A New Tabu Search Heuristic for the Open Vehicle Routing Problem

Z Fu et al--Tabu search heuristic for OVRP 269

A non-negative cost, ci.,

is associated with each edge (i, j) EE and represents the travel cost spent to go from vertex i to vertex j. A complete network is assumed. If any edges between vertices are missing in the original graph, then they are replaced by edges with artificial high costs.

Each customer i (i= 1, ..., n) is associated with a known

non-negative demand, di, to be delivered or picked up, and the depot has a fictitious demand do = 0. Given a vertex set SC V, let d(S)= Eisdi denote the total demand of the set.

A set of identical vehicles, each with capacity C, is available at the depot. To ensure feasibility we assume that di < C for each i= 1, ..., n. Each vehicle may perform at most one route. Given a set S C V\{0}, we denote by Kmin the minimum number of vehicles needed to serve all customers in S. Often, Kmin

is replaced by the lower bound

[d(S)/C]. Then let Cavg- [d(S)/Kmin ] be the correspond-

ing upper bound of average vehicle load. When S = V\{0}, then

Kmin is written as Kmin and Cs, is written as Cavg. The

maximum route length is L if such a constraint is included.

The tabu search heuristic

Tabu Search (TS) is a heuristic method designed to guide other methods, including local search algorithms, to escape local optima. It was first introduced by Fred Glover in 1986 and has since been used to solve many practical applications. A thorough discussion may be found in Glover and Laguna.7

TS introduces a tabu list to forbid certain moves which would allow the search to return to a previous solution and become trapped in a local optimum. For example, if the current solution is at a local optimum, the next solution to be accepted would be one that gives the least increase in the objective. Without a tabu restriction to prevent a reverse of this move, it is possible that at the next iteration the best move available would be to go back to the local optimum and the search would be trapped.

The basic idea of TS is the following. Begin with an initial solution. At each iteration, a neighbourhood of the current solution is created through different classes of moves. Then, the best admissible solution in the neighbourhood is selected as the new current solution, and the procedure is repeated until a stopping criterion is satisfied. A move is always admissible if it is not tabu. If it is tabu, then it may still be admissible if the move produces a solution strictly better than the best solution so far. In both cases, the accepted move is recorded, and the tabu restrictions are updated.

Over the last 10 years, several metaheuristics have been proposed for the VRP, such as simulated annealing, tabu search, genetic algorithms, ant algorithms, and neural net- works. A comprehensive survey is provided by Gendreau et al and Cordeau et al.9 According to their analyses, in general, TS emerges as the most effective approach for the VRP so far, both in solution quality and computing time. Furthermore, there has been a tendency toward the

development of faster, simpler (with fewer parameters), and more robust algorithms. Cordeau et a19 also propose four criteria to measure good heuristics, which are accuracy, speed, simplicity and flexibility. Based on this guidance, the following TS heuristic is proposed for the OVRP.

Initial solution

An initial solution is required for any TS algorithm to start the local search process. With respect to the TS metaheur- istic for the VRP, Van Breedam'o states recently that '... the performance of the ... TS heuristic is highly dependent on the quality of the initial solutions.' Brandio3 also shows in his recent research that the initial solution can give an important contribution to enhance the final solution, and uses two methods for obtaining the initial solutions: a nearest neighbour heuristic and a pseudolower bound.

In this paper, for the initial solution, besides generating randomly, by exploiting the problem structure, we propose an optimization-based farthestfirst heuristic (FFH) for it. A good heuristic approach is usually one that can carefully exploit the problem structure. In the OVRP, because the vehicle routes are open ones, it is often the case that in the best solutions, the farthest customer from the depot is at the end of a route. Based on this observation, in the proposed heuristic, new routes are always formed from the farthest unrouted customer from the depot.

The basic idea of the FFH is the following. A new route starts from the farthest unrouted customer i from the depot. Along the shortest route, back from i to the depot, add customers to this new route until the vehicle is full enough. If the vehicle is not sufficiently full, the route is rejected and the next shortest route is considered instead. The process is repeated until a route is accepted. The procedure is described in more detail below.

Step 0: Initialisation. Find the shortest distance from the depot to every customer.

Step 1: Starting new route from the farthest unrouted customer i. Choose the farthest customer i whose demand is not assigned to any vehicle.

Step 2: Adding other customers to this new route until the vehicle is full enough. Set k = 0; repeat

increment k; testing: along the kth shortest route from i back

to the depot, add those customers j on this route one by one to this vehicle so that both (di +

-jdj C) and (total length of the

route 4 L) hold, until no more can be added; if [(vehicle capacity left<min {(any unassigned

d,), (C-Cavg)}) or (no demands left)] then accept the route;

until a route is accepted.

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270 Journal of the Operational Research Society Vol. 56, No. 3

Step 3: Accept the new route. Update the data accordingly; if all customers are routed then end of the FFH else go to Step 1.

The strategy of not starting a new route until the vehicle currently considered is full enough aims for a solution that keeps the number of vehicles needed to a minimum. Furthermore, selecting the shortest, second shortest,..., kth shortest route from customer i back to the depot to test in turn attempts to minimise the total travelling cost. The procedure due to Lawler" can be used to find the kth shortest route from customer i back to the depot. However, the FFH heuristic may be slow if step 2 needs to be repeated many times until a route is accepted.

The FFH is based on the heuristics in our previous case studies,4,5 and with some necessary modifications. The idea of always starting new route from the farthest unrouted customer in the FFH seems to perform well not only in node

routing problems, but a similar idea is also applied to arc routing problems by Li and Eglese,'2 in which 'At the

beginning of the construction of each route, the farthest ungritted arc is chosen to be treated on the route'.

Neighbourhood structure

Various neighbourhood structures have been presented for the application of TS to the VRP by several authors. Pureza and Franga'3 defined the neighbours of a solution by moving a vertex to a different route or by swapping vertices between two routes while preserving feasibility. In Osman,'4 neighbourhoods are defined by means of the 2-interchange generation mechanism, with = 2. This includes a combina- tion of 2-opt moves, vertex reassignments to different routes, and vertex interchanges between two routes. The neighbour- hood structure proposed by Gendreau et al'5 is defined by removing a vertex from its current route and inserting it into another route containing one of its p nearest neighbours using GENI, a Generalized Insertion procedure developed by them for the TSP. Taillard et a116 introduced another

exchange heuristic, which is called the CROSS exchange, to

generalize two edge exchange heuristics for problems with time windows. In a TS heuristic for the vehicle routing problem with backhauls and time windows, Duhamel et a1'7 presented a stochastic mechanism in which the current

neighbourhood is randomly selected among 2-opt, Or-opt, and swap. Recently, Cordeau et a1'8 introduced yet another neighbourhood structure.

With respect to the OVRP, Brandio3 defines the neighbourhood by two types of trial moves, insert and swap. An insert move consists of taking a customer from one route and inserting it into another route, and a swap move consists of exchanging two customers belonging to two different routes.

In our implementation, different classes of neighbour- hood moves are applied to the current solution. These moves

are based on the 2-interchange generation mechanism but with a combination of vertex reassignment, vertex swap, 2-opt and 'tails' swap, within the same route or between two routes. Some of the possible transformations can be shown in the following, where the selected vertices are underlined.

Select two different vertices (customer or depot, within the same route or different ones) randomly. Perform one of the following four moves randomly:

(1) Vertex Reassignment. Remove the first selected vertex from its current position on the route and insert it into the position before (after-for pickup problem) the second selected vertex, that is, X -

(013560479028)--- X2 (015604790238), X, -(013506470928) -+X2- (013564700928).

(2) Vertex Swap. Exchange the positions of two selected vertices, that is, XI = (013506479028) -+ X2 - (013546079028), X1 - (013560479028) -)X2 = (01350 0479628).

(3) 2-opt. Reverse the order of all elements between two selected vertices like the standard 2-opt move in TSP, if two selected vertices are within the same route, that is, Xl = (013564079028) AX2 (014653079028), Xl = (013560479028) -+ X2 - (013569740028).

(4) 'Tails' Swap. Exchange the 'tails' after two selected vertices (from the selected vertex to the end of the route; both vertices must be customers), if two selected vertices are in different routes, that is, XI= (013560479028) X2 -(013790456028).

Note that (a)-(c) are the moves that can possibly reduce the number of routes. Furthermore, the first 0 (or the last 0) of a solution for the delivery (or pickup) problem is not allowed to be selected and removed during the 2-interchange process.

This neighbourhood structure is one that allows moves to infeasible solutions in terms of the vehicle capacity or route

length. This structure is able to enhance the TS mechanism and may result in better improvement on both travelling costs saving and eliminating an existing route.

Evaluation of solutions

The OVRP has two objectives to be optimized: total

travelling cost and operating cost (the number of vehicle used). As mentioned in the Introduction, the priority is given to the operating cost. Therefore, a feasible solution with a certain number of vehicles always dominates over any other feasible solutions requiring more vehicles. For those solu- tions with the same number of vehicles required, the one with minimum total travelling cost is selected.

To facilitate the exploration of the search space, a move is allowed even if it results in an infeasible solution. The extent of the infeasibility can be measured by incorporating the vehicle capacity and maximum route length constraints into

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Z Fu et a/-Tabu search heuristic for OVRP 271

the objective function by adding a penalty if the constraints are broken. We use the same equation as Branddo's,3 that is,

K

I[T(r) + p(Ec(r) + Et(r))] r=l

where K is the total number of routes in the solution, T(r) is the travelling cost of route r, Ec(r) and Et(r) are the excess of load and duration in route r, respectively, and p is the penalty coefficient. Ec(r) and Et(r) are equal to zero for all routes if a solution is feasible. Pe [0.00001, 200000] is initially equal to 1 and weighted by a self-adjusting parameter: every 10 iterations, it is divided by 2 if all 10 previous solutions were feasible or multiplied by 2 if all were infeasible. This mechanism, which was used by Gendreau et a115 in the Taburoute algorithm for the VRP by introducing two independent penalties, produces a mix of feasible and infeasible solutions and lessens the likelihood of being trapped in a local minimum.

Tabu list

The tabu list contains the move attributes of solutions during the last 5-10 (selected randomly) iterations. A set of (n + 1) x (n + 1) matrices can be constructed for the record of tabu status. If vertex i is selected for the move (a), Vertex Reassignment, the tabu status is saved in the element (i, i) of a matrix. If vertices i and j are selected for the moves (b)-(d), the tabu status is saved in the element (i, j) of a matrix. At each iteration, the tabu status of the last move performed is added to the list, while the others are decreased by one until equal to zero.

Stopping criterion

The search is terminated if either a specified number of iterations has elapsed in total or since the last best solution was found. The following variables are used in the description of the TS heuristic:

iter current number of iterations; max_iter maximum number of iterations; cons iter current number of consecutive iterations without any improvement to the best solution so far; max cons iter maximum number of consecutive iterations without any improvement to the best solution so far; cand_list current number of candidate moves on the list; max cand list maximum number of candidate moves on the list.

TS heuristic

Generate an initial feasible solution randomly or by the FFH, and set this solution as the current solution and the best solution so far;

Set iter and cons_iter to 0; While (iter < max_iter) and (cons_iter < max_cons_iter) do Begin

While (cand list < max_cand list) do begin

Select two vertices randomly; Select one of the four types of neighbourhood

move randomly; Add the solution produced by the selected move to

the candidate list; end; Select the best solution in the candidate list if it is not

tabu, or it produces a solution strictly better than the best solution so far;

Set the new solution as the current solution, update the tabu list and increment iter;

If the new solution improves the best solution so far, update the best solution so far, and set consiter to 0; Otherwise, increment consiter;

End.

The distinctive feature of this TS heuristic is the use of a simple but powerful neighbourhood structure. During the search, the current neighbourhood is randomly selected among four types of neighbourhood move, and the tabu length is randomly selected between 5-10. This mechanism introduces a stochastic type of diversification. Furthermore, by applying a different transformation at each iteration, a larger neighbourhood is explored over a few iterations, without the computational burden associated with an extensive search in a unified neighbourhood. The use of the penalty function approach allows the search process to examine solutions that may be infeasible with respect to the capacity and duration constraints.

Computational results and comparison

This TS heuristic has been programmed using Turbo Pascal and implemented on a Pentium-II PC running at 600 MHz with 128 MB RAM. To test the computational performance of the heuristic, we compared it on problems taken from the literature with Sariklis and Powell's2 algorithm, and

Brand.o's3 algorithm. To show the influence of the initial

solution on final solutions from the TS algorithm for the OVRP, we also compared the performance resulting from initial solutions generated by the FFH with ones generated randomly.

The results were produced with the following bounds: max_cand_list = 150 + 2n, maxcons_iter - 4000 + 10 On, and max iter = 12000 + 30n.

Comparison of the heuristic on problems taken from the literature with two reported algorithms

The test problems are those provided by Christofides et a119 (can be downloaded from the OR Library at address

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272 Journal of the Operational Research Society Vol. 56, No. 3

http://mscmga.ms.ic.ac.uk/jeb/orlib) and Fisher,20 denoted by prefix letter C and F, respectively. The characteristics of the problems and the results are presented in Table 1. All these problems are Euclidean and it is assumed that the distance between each pair of customers is equal to the

travelling cost. For the problems with a route length constraint, the maximum route length (L) is equal to the

original route length limit for the corresponding VRP

multiplied by 0.9 (this transformation is required because the original problems have been conceived for the VRP and not for the OVRP); furthermore, there is a service time

(droptime) of each customer, but in the solution we report only the travelling cost, as in Brandio.3 The number of vehicles required (K) is given if it is different from Kmin.

Brandio (C) and (D), denoted in his paper, correspond to the final solutions produced by the two initial solution methods, respectively. CPU is the computing time in seconds. Some problems are without route length constraint, and some test problems are not solved by Sariklis and Powell.2

According to the initial solutions that are generated randomly or by the FFH, our results are shown in column

(R) or (F), respectively. For each problem, the results are the best solutions from among 20 runs. They show that the final solutions found for the test problems by our TS heuristic are better in six cases out of seven than Sariklis and Powell's method;2 and better than Brandio's TS

algorithm3 in nine cases out of 16, and are the best solutions so far (see bold values in Table 1). But note that Sariklis and Powell's times are considerably less than ours and Bran- dao's. A method capable of quickly providing a good- quality solution can be important in some cases.

Comparison of the influence of the initial solution on final solutions

To show the influence of the initial solution on final solutions, the average cost, standard deviation of cost for the final solution, and the average CPU time of 20 runs for each test problem (for the same number of K) with the initial solutions generated randomly (Ran) and by the FFH, respectively, are given in Table 2. For the initial and final solutions, the average cost saving of the FFH from 'Ran' is listed in columns 'Ini_save' and 'Fin_save', respectively, while the corresponding extra CPU time required is in column 'Fin more'. On average, the initial solutions

generated by the FFH have 54.1% of travelling cost saving, but only result in 0.3% travelling cost saving on final solutions in comparison with those generated randomly; and 325.5% more CPU time is required. The test data are based on a complete network. It is expected that the FFH

algorithm will be more efficient when the original network is not complete and corresponds to a road or rail network.

Conclusions

OVRPs exist in distribution management, as well as in rail, bus and air transportation. In this paper we proposed a TS heuristic for the OVRP with vehicle capacity and route length constraints. Starting with the initial solutions generated randomly and by the proposed farthest first heuristic (FFH) respectively, the final solutions, in nine cases out of 16, found

by our TS heuristic are better than those in the literature in that fewer vehicles are required. By comparing the perfor- mance of the TS heuristic starting with the initial solutions

Table 1 Comparison of algorithms on literature problems

Sariklis & Powell Branddo (C) Branddo (D) Fu, Eglese & Li (R) Fu, Eglese & Li (F)

Problem n L Kmin (K)/Cost CPU* (K)/Cost CPUt (K)/Cost CPUt (K)/Cost CPU_

(K)/Cost CPUt

C1 50 5 488.2 0.22 438.2 1.7 416.1 88.8 413.3 65.0 408.5 170.2 C2 75 10 795.3 0.16 584.7 4.9 574.5 167.5 570.6 197.8 587.8 202.1 C3 100 8 815.0 0.94 643.4 12.3 641.6 325.3 617.0 367.6 644.3 719.9 C4 150 12 1034.1 0.88 767.4 33.2 740.8 870.2 741.1 1094.0 734.5 1610.3 C5 199 16 1349.7 2.20 1010.9 116.9 953.4 1415.0 (17)/886.6 1279.9 (17)/878.0 2060.5 C6 50 180 5 (6)/416.0 1.4 (6)/413.0 55.8 (6)/409.7 184.9 (6)/400.6 128.0 C7 75 144 10 (11)/581.0 3.4 634.5 123.7 (11)/560.4 231.8 (11)/565.7 292.4 C8 100 207 8 (9)/652.1 10.4 (9)/644.6 351.7 (9)/647.7 321.0 (9)/638.2 987.8 C9 150 180 12 (14)/827.6 25.2 (13)/785.2 622.2 (14)/752.0 799.5 (14)/758.9 1635.2

C10 199 180 16 (17)/946.8 100.1 (17)/884.6 2060.3 (18)/898.2 1218.2 (18)/891.3 1922.2 ClI 120 7 828.3 1.54 713.3 15.7 683.4 696.0 716.5 88.9 753.8 735.8 C12 100 10 882.3 0.76 543.2 7.8 535.1 233.6 534.8 30.9 549.9 413.4 C13 120 648 7 (11)/994.3 25.8 (11)/943.7 401.9 (12)/952.4 35.0 (12)943.0 741.1 C14 100 936 10 (12)/651.9 8.1 (11)/597.3 419.8 (12)/469.3 155.5 (12)586.8 463.2

Fll 71 4 179.5 5.7 177.4 398.1 175.0 151.7 178.0 256.0 F12 134 7 825.9 32.7 781.2 1000.2 778.5 125.4 789.7 1044.8

*Seconds on a Pentium at 133 MHz with 16 MB RAM. tSeconds on a Pentium III HP Vectra VEi8 at 500 MHz. tSeconds on a Pentium II at 600 MHz with 128 MB RAM.

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Page 8: A New Tabu Search Heuristic for the Open Vehicle Routing Problem

Table 2 Comparison of the influence of initial solution on final solutions

Random initial solution FFH initial solution

Average cost Average CPU time Average cost Average CPU time

Standard Ini save Fin save Standard Fin more Problem n L Kmin Initial Final deviation of cost Initial Final Initial (%) Final (%) deviation of cost Initial Final (%)

Cl 50 5 1565.9 425.5 11.89 0.0 76.1 706.4 54.9 419.9 1.3 7.73 105.0 195.9 157.4 C2 75 10 2449.2 606.1 22.99 0.0 151.5 760.2 69.0 615.2 -1.5 19.45 65.4 178.4 17.8 C3 100 8 3296.7 652.7 11.10 0.0 220.9 1322.5 59.9 655.0 -0.4 8.71 417.3 605.0 173.9 C4 150 12 4827.5 773.2 29.32 0.0 429.4 1465.0 69.7 765.0 1.1 16.90 800.9 1207.3 181.2 C5 199 16 6339.1 933.5 44.17 0.0 414.5 2129.8 66.4 904.6 3.1 19.38 483.1 1349.0 225.5 C6 50 180 5 1966.2 414.3 3.66 0.0 90.9 1151.6 41.4 411.4 0.7 4.61 76.8 128.7 41.6 C7 75 144 10 2888.5 576.3 12.83 0.0 171.5 1507.8 47.8 575.9 0.1 9.14 32.1 200.0 16.6 C8 100 207 8 4028.4 656.6 10.61 0.0 319.5 2211.6 45.1 650.8 0.9 4.71 398.4 688.1 115.4 C9 150 180 12 5921.8 769.0 9.65 0.0 620.3 3079.2 48.0 774.1 -0.7 11.42 808.1 1345.4 116.9 C10O 199 180 16 7741.1 912.9 11.68 0.0 760.7 4072.3 47.4 915.2 -0.3 16.16 433.5 960.9 26.3 Cli 120 7 6344.5 842.3 56.71 0.0 64.0 2459.8 61.2 817.8 2.9 67.33 539.1 603.3 842.7 C12 100 10 3776.6 639.2 86.51 0.0 26.0 1291.8 65.8 599.3 6.2 47.99 399.1 424.2 1531.5 C13 120 648 7 12000.9 999.6 34.68 0.0 138.3 8459.8 29.5 1042.8 -4.3 59.02 542.3 600.8 334.4 C14 100 936 10 12717.1 626.9 54.36 0.0 66.3 10158.0 20.1 624.4 0.4 33.17 324.7 391.2 490.0 Fll 71 4 1245.8 200.4 21.21 0.0 36.3 396.3 68.2 196.0 2.2 13.80 152.2 179.4 394.2 F12 134 7 6480.9 861.8 50.34 0.0 103.6 1902.2 70.6 919.4 -6.7 77.98 602.5 665.2 542.1

Mean 54.1 0.3 325.5

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Page 9: A New Tabu Search Heuristic for the Open Vehicle Routing Problem

274 Journal of the Operational Research Society Vol. 56, No. 3

generated by the special heuristic (FFH) with ones generated randomly, it shows that there is not much influence of initial solution on final solutions in this TS heuristic for the OVRP for the test data based on a complete network.

Further research work may include looking at OVRPs with other constraints, such as time windows, and for the reason the number of vehicles required in some cases found

by our algorithm is not the minimum.

Acknowledgements-This research was supported by National Natural Science Foundation of China (NSFC, 70071003) and China Scholar- ship Council. We were also grateful to Jose Brandio and the two anonymous referees whose comments helped us to improve the presentation of this paper.

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Received January 2003; accepted May 2004 after three revisions

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